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OF THE 






STEAM-ENGINE. 



CONTAINING ALL THE RULES REQUIRED FOR THE RIGHT 

CONSTRUCTION AND MANAGEMENT OF ENGINES OF EVERY CLASS, WITU 

THE EASY ARITHMETICAL SOLUTION OF THOSE RULES. 



CONSTITUTING 



A KEY 



CATECHISM OF THE STEAM-ENGINE.' 



ILLUSTRATED BY 

SIXTY-SEVEN WOOD-CUTS, AND NUMEROUS TABLES AND EXAMPLES. 




BY 



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1 



JOHN BOURNE, C. 

'a treatise on the steam-engine, 1 *a teeatise 

-PKOPELLEK. 1 \a CATECHISM OF THE STjEAM-ENGINE, 1 ETcT^ 





pq I 

. NEW YORK : 
AEMTON AND C 0<jtfP Y N Y > 
54$£& 5 51 BROADWiA 
1874. 







By irausfer 

OCT 11 1916 



,w 




? ^ 



GEOEGE TUMBULL, ESQ., C.E., F.E.A.S., ETC., 



LATE ENGINEER-IN-CHIEF OF THE EAST INDIAN RAILWAY. 



My dear Me. Ttjbnbull, 

In dedicating the present Work to you I am moved by 
two main considerations : — First, to testify in the best manner 
I can my regard and esteem for you personally ; and Second, to 
mark my sense of the skill, tact, and abiding integrity which 
you brought to the onerous duty of constructing the first and 
greatest of the Indian railways, and of which, while in India, I 
had opportunities of forming a just appreciation. 

The public in this country — traditionally so ignorant of 
India — has yet to learn the important fact, that the works 
carried out under your direction in that country, are greater 
and more difficult than most of those which are to be found at 
home; and that among other achievements, you constructed 
the largest bridge in the world — the great bridge over the St. 
Lawrence alone excepted. But these technical successes, im- 
portant as they are, were not more eminent than those which 
you won over the discouragements and difficulties of the Indian 
official system — ending, too, in gaining the esteem and appro* 



IV DEDICATION. 

bation of the Indian Government, as well as of those for whom 
yon zealously labored for so many years in India. 

"Whatever the benefits may be of the Indian railways, their 
greatest benefit is that they have taken to that country men who 
have impressed the people with their skill, and who have ac- 
quired an accurate perception of the physical wants of the 
country, together with all that practical knowledge of localities 
which will enable them to carry out with confidence, economy, 
and success, the numerous improvements still required by that 
great dependency, and upon which only a comparatively small 
beginning has yet been made. 

I remain, my dear Mr. Turnbull, 

Truly yours, 

J. BOUENE. 



PREFACE 



The present work, designed mainly as a Key to my 
* Catechism of the Steam-Engine/ has, during its compo- 
sition, been somewhat extended in its scope and objects, so 
as also to supply any points of information in which it 
appeared to me the Catechism was deficient, or whereby 
the utility of this Handbook as a companion volume would 
be increased. 

The purpose of the Catechism being rather to enun- 
ciate sound principles than to exemplify the application of 
those principles to practice, it was always obvious to me 
that another work which would point out in the plainest 
possible manner the methods of procedure by which all 
computations connected with the steam-engine were to be 
performed — illustrated by practical examples of the appli- 
cation of the several rules — was indispensable to satisfy 



VI PREFACE. 

the wants of the practical engineer in this department 
of enquiry. The present work was consequently begun, 
and part of it was printed, several years ago, but the 
pressure of other pursuits has heretofore hindered its 
completion ; and in now sending it forth I do so with the 
conviction that I have spared no pains to render it as 
useful as possible to the large class of imperfectly educated 
engineers to whom it is chiefly addressed. It is with the 
view of enabling its expositions to be followed by those 
even of the most slender scientific attainments that I have 
introduced the first chapter, explaining those several pro- 
cesses of arithmetic by which engineering computations 
are worked out. For although there is no want of man- 
uals imparting this information, there are none of them, 
that I know of, which have special reference to the wants 
of the engineer ; and none of them deal with those asso- 
ciations, by way of illustration, with which the engineer 
is most familiar. Indeed, engineers, like sailors and other 
large classes of men, have an order of ideas, and, to some 
extent, even a species of phraseology of their OAvn ; and 
the avenues to their apprehension are most readily opened 
by illustrations based upon their existing knowledge and 
experience, such as an engineer can best supply. By this 
familiar method of exposition the idea of difficulty is dis- 



PREFACE. Vll 

pelled ; and science loses half its terrors by losing all its 
mystery. 

If I might infer the probable reception of the present 
work from the numerous anxious enquiries addressed to 
me from all quarters of the world during the last ten 
years, touching the prospects of its speedy appearance, I 
should augur for it a wider popularity than any work I 
have yet written. The questions propounded to me by 
engineers and others, in consequence of the offer I made 
in the preface to my ' Catechism of the Steam-Engme,' in 
1856, to endeavour by my explanations to remove such 
difficulties as impeded their progress, have had the effect 
of showing more clearly than I could otherwise have per- 
ceived what the prevalent difficulties of learners have 
oeen ; and I have consequently been enabled to give such 
explanations in the present work as appeared best calcu- 
lated to meet those difficulties for the future. 

To several of my correspondents I have to acknowledge 
myself indebted for the correction of typographical errors 
in my several works, and also for valuable suggestions of 
various kinds, which I have made use of in every case in 
which they were available. 

I may here take occasion to notify that I have lately 
prepared an Introduction to my * Catechism of the Steam- 



yjj! PKEFACE. 

Engine/ which reviews the most important improvement? 

of the last ten years ; and which, for the convenience of 

persons already possessing the Catechism, may be had 

separately. These three works taken together form a body 

of engineering information so elementary as to be intelli 

gible by anybody, and yet so full that the attentive student 

of them will, I trust, be found not to fall far short of the 

most proficient engineers in all that relates to a knowledge 

of the steam-engine in its most important applications. 

J. BOUBNE. 

Berkeley Yilla, Regent's Park Road, 
London : 1865. 




CONTENTS. 



CHAPTER I. 

ARITHMETIC OP THE STEAM-ENGINE, 



Pr'.nciples of Numeration 

Addition ...... 

Subtraction ..... 

Multiplication . 

Division ...*.. 

Nature and Properties of Fractions 

Addition and Subtraction of Fractions 

To Reduce Fractions to a Common Denominator 

Multiplication and Division of Fractions 

Proportion, or Rule of Three 

Squares and Square Roots of Numbers 

Cubes and Cube Roots of Numbers . . 

On Powers and Roots in General 

Roots as represented by Fractional Exponents . 

Logarithms ..... 

Compound Quantities . . . . 

Resolution of Fractions into Infinite Series 

Equations ....... 



PAQB 
1 

10 
13 
16 
24 
30 
34 
35 
38 
42 
44 
48 
49 
51 
52 
57 
G6 

n 



CHAPTER II. 

MECHANICAL PRINCIPLES OF THE STEAM-ENGINE. 



Law of the Conservation of Force 
Law of Virtual Yelocities . 
Nature of Mechanical Power . 



78 
79 
90 



X COSIES iz. 






PAG* 


Mechanical Equivalent of Heat. .... 


SI 


Laws of Felling Bodies ..... 


. 93 


Motion of Fluids ...... 


100 




. IC-5 


Centrifugal Force 


11 7 


Bodies ReTolrini: in a Circle ..... 


. ::; 


Centres of Gyration and Percussion 


112 


The Pendulum 


. 114 


T„e G"'OTernoT 


116 


Friction ........ 


. 113 


S:rength of Materials ..... 


124 


Strength of Pillars.. Beams, and Shafts 


. 12-8 


CHAPTER III. 




THEORY OF THE STEAM-HSGIXE. 




Bstnre and Effects of Heat . 


. 134 


Diffe: :_:-:' ~: ;•: 1 :..-_.: ;.:a:e and Quantkv of Heat 


136 


Abs :lute Zero ....... 


. 136 


Tin: 1 Tomt -:: :.:v.: :-s 


137 


Thermometers ....... 


. 1ST 


Dilatation ....... 


140 


Liquefaction ....... 


. 150 


Vaporisation ...... 


152 


Pressure of Steam at Different Temperatures 


. 157 


Specific Heat ...... 


162 


Phenrmena of Elulii:i:n . 


. 168 


Communication of Heat ..... 


171 


Combustion ... ... 


. 174 


Thermodynamics ...... 


180 


Ext:.ns::n of Steam 


. 1S2 


V-.".: ;itv and Friction of Funning Water 


199 



Nominal Power 
G meral Proper 
Steam Ports 
Steam Pipe . 
Safety Valves 
Feed Pipe 



CHAPTER IV. 
??. ~?:r.r::'\"S o? steam -ex GENtES. 



213 
212 
216 
218 
219 
22] 



CONTENTS. 


XI 




PAGH 


Air-Pump and Condenser .... 


. 222 


Injection Pipe ...... 


222 


Foot Valve . ... 


. 223 


Feed Pump .....•• 


224 


Land Engines : — Cold "Water Pump . 


. 227 


Fly Wheel . 


228 


Governor .... 


. 230 


Piston Rod . . . . 


231 


Main Links ... 


. 232 


Air-Pump Rod . 


232 


Back Links , . . 


. 232 


Studs of the Beam . • • 


232 


Main Centre . . . 


. 232 


Main Beam . . . 


233 


Connecting Rod ... 


. 238 


Fly-wheel Shaft 


238 


Crank . . . • 


. 240 


Crank-Pin . . . . 


246 


Mill Gearing 


. 246 


Marine Engines ; — Crosshead . . . 


254 


Side Rods . . . 


. 258 


Piston Rod . . . . 


261 


Connecting Rod . . . 


. 263 


Crosstail , 


267 


Side Lever and Centres 


. 267 


Crank . . . < 


271 


Paddle Shaft 


. 278 


Air-Pump 


280 


Air-Pump Rod . . . 


. 280 


Air-Pump Crosshead 


282 


Air-Pump Side Rods 


. 284 


Dimensions of Caird and Co.'s Marine Engines 


287 


Dimensions of Maudslay and Field's Marine Engines 


. 290 


Dimensions of Seaward and Co.'s Marine Engines 


292 


Tables of Proportions of Engines 


. 294 


Locomotive Engines .... 


301 



CHAPTER V. 

PROPORTIONS OF STEA1I-BOILERS. 



Velocity of Draught in Chimneys 
Proportions of Wagon Boilers 



503 

no 



xii coxte:s-ts %•. 

PAoa 

Proportions of Rue Boilers ...... 311 

Proportions of Modern Boilers . . . 314 

Indications to be fulfilled in Marine Boilers . . . 316 

Strength of Boilers . . . . 320 

Example of a Locomotive Boiler . ... 329 

CHAPTER VI. 

POWEE AND PEEP0E2IANCE OF ENGINES, 

Construction and use of the Indicator .... 333 

Counter, Dynamometer, and Duty Meter . . . S72 

Heating Surface in Modern Boilers ..... 375 

Relative Surface Areas in Boilers and Condensers . . 3S0 

Giffard's Injector ....... 383 

Comparative Efficacy of Hydraulic Machines . . . 385 

Power required to Drive different Factories . . . 3SG 

CHAPTER TH. 

STEAM NAVIGATION. 

Resistance of Vessels ...... 399 

Friction of Water ....... 4>3 

Speed of Steamers of a Given Power . . . .429 

General Conclusions ...... 453 

Examples of Lines of Approved Steamers .... 454 







RECEIVED. ^« 



^/P^A^I^ 



; '-.» 




HANDBOOK 



OF 



THE STEAM-ENGINE 



CHAPTER I. 



AEITHMETIC OF THE STEAM-ENGINE. 



In this chapter I propose to explain as plainly and simply as I 
can those principles of arithmetic which it is necessary to know, 
that we may be able to perform all ordinary engineering calcula- 
tions. In order that my remarks may be generally useful to work- 
ing mechanics of little education, I shall proceed upon the suppo- 
sition that the reader is not merely destitute of all arithmetical 
knowledge, but that he has no ideas of number or quantity that 
are not of the most vague and indefinite description. I have known 
many engineers — who were otherwise men of ability — to be in 
thi3 condition ; and the design of these observations is to enable 
such, with the aid of their own common sense and their familiar 
associations, to arrive at tangible ideas respecting the properties of 
numbers, and to perform with facility all the ordinary engineering 
calculations which occur in the requirements of engineering prac- 
tice. These various topics are not beset with any serious diffi- 
culty. The processes of arithmetic are merely expedients for faci- 
litating the discovery of results which every mechanic of ordinary 
ingenuity would find a means of discovering for himself, if 
really called upon to set about the task ; and it is mainly be- 
1 



2 ARITHMETIC OF THE STEAM-ENGINE. 

cause the rationale of these processes has not "been much ex- 
plained in school treatises, but the results presented as feats of 
legerdemain performed by the application of a certain rule — the 
reason of which is not made apparent — that the idea of difficulty 
has arisen in connection with such enquiries. The rudest and 
most savage nations have all some species or other of arithmetic 
suited to their requirements. The natives of Madagascar, when 
they wish to count the number of men in their army, cause the 
men to proceed through a narrow pass, where they deposit a 
stone for each man that goes through; and by subsequently ar- 
ranging these stones in groups of ten each, and these again in 
groups of a hundred, and so on, they are enabled to arrive at a 
precise idea of the number of men the army contains. A la- 
bourer in counting bricks out of a cart or barge makes a chalk- 
mark on a board for every ten bricks he hands out ; and these 
chalk-marks he arranges in groups of five or ten each, so that 
he may easily reckon up the total number of groups the board 
contains. These are expedients of numeration which the most 
moderate intelligence will suggest as conducive to the acquisi- 
tion of the idea of quantity; and the rules of arithmetic are 
merely an extension and combination of such methods as expe- 
rience has shown to be the most convenient in practice to ac- 
complish the ends sought. 

It will be obvious that the number of stones or chalk-marks 
collected into groups in the preceding examples may either be 
five, ten, twelve, or any other number ; the only necessary con- 
dition being that the number in each group shall be the same. 
The concurrent practice of most nations, however, is to employ 
groups consisting of ten objects in each group ; no doubt from 
the circumstances that mankind are furnished with ten fingers, 
and because the fingers are much used in most primitive systems 
of numeration. In some cases, however, objects are reckoned 
by the dozen, or score, or gross ; or, in other words, a dozen, a 
score, or a gross 'of objects are collected in each group. But in 
the ordinary or decimal system of numeration, ten objects or 
units are supposed to be collected in each group, and ten of these 
primary groups are supposed to be collected in each higher or 



DIFFERENT EXPEDIENTS OF NUMERATION. 3 

larger group of the class immediately above, and so on indefi- 
nitely. The decimal system is so called from the Latin word 
decern, signifying ten, and the word unit is derived from the 
Latin word units, signifying one. Ten units form a group of ten, 
and ten of these groups form a group of a hundred, and ten groups 
of a hundred form a group of a tliousand, and so on for ever. 

The Romans, whose numbers are still commonly used on clock 
faces, employed a mark or i to signify one ; two marks or n to 
signify two ; three marks or in to signify three ; and four marks 
or mi to signify four. But as it would have been difficult to 
count these marks if they became very numerous, they employed 
the letter v to signify five and the letter x or a cross to signify 
ten, and v is the same mark as one-half of x, which was no doubt 
the primary of the two characters. An i appended to the left- 
hand side of the v or x signified v or x diminished by one, 
whereas each additional i added to the right-hand side of the v 
or x, signified one added to v or x. Thus according to the 
Eoman numeration iv signifies four ; vi signifies six ; ix signifies 
nine ; xi signifies eleven ; xn signifies twelve ; and so on. A 
hundred is signified by the letter c, the initial letter of the Latin 
word centum, signifying a hundred; and a thousand is repre- 
sented by the letter m, the initial letter of the Latin word mille, 
signifying a tliousand. 

It is clear that the Eoman numeration, though adequate to the 
wants of a primitive people, was a very crude and imperfect sys- 
tem. It has therefore been long superseded for all arithmetical 
purposes by the system of notation at present in common use, 
and which has a distinct sign or figure for each number up to 9, 
and a cipher or 0, which has no individual value, but which af- 
fects the value of other figures. This system, which came origi- 
nally from India, was brought into Europe by the Moors ; and in 
common with most of the oriental languages, it is written from 
right to left instead of from left to right, like the languages of 
Europe, so that in performing a sum in arithmetic — as in writing 
a word in Sanscrit or Arabic — we have to begin at the right- 
hand side of the page. In this system the classes or orders of 
the objects or groups of objects is indicated by the place occu- 



4 ARITHMETIC OF THE STEAM-ENGINE. 

pied by the figures which express their value. Thus iu the case 
of the groups of stones employed in Madagascar, the figure S 
may be employed to designate either three individual stones, or 
three groups of ten each, or three groups of a hundred each ; but 
in using the figure it is quite indispensable that it should appear, 
by some distinctive mark, which order or class is intended to be 
designated. "We might use the figure 3 to designate three single 
stones, and we might use the figure with a circle round it to de- 
note groups of ten each, and with a square round it to denote 
groups of a hundred each. But on trial of such a system we 
should find it to be very cumbrous and perplexing, and the method 
found to be most convenient is to add a cipher after the three to 
show that groups of tens are intended to be signified, and two 
ciphers to show that groups of hundreds are intended to be sig- 
nified. Three groups of tens, or thirty, are therefore expressed 
by 30, and three groups of hundreds are expressed by 300. 
Here the ciphers operate wholly in advancing the 3 into a higher 
and higher position, which, however, other figures will equally 
suffice to do if there are any such to be expressed. Three groups 
of one hundred stones in each, three groups of ten stones in each, 
and three individual stones, will therefore be represented by the 
number 333, in which the same figure recurs three times, but 
which is counted ten times greater at each successive place to 
which it is advanced, reckoning from the right to the left. Of 
course, the number three hundred and thirty-three might be 
represented in an infinite number of other ways, differing more 
or less from the one here indicated ; and any of the properties 
belonging to the number would equally hold by whatever 
expedient of notation it was expressed. But the manner here 
described is that which the accumulated experience of mankind 
has shown to be the most convenient; and it is therefore gen- 
erally adopted, though it is proper to understand that there is no 
*nore necessary relation between the number itself and the com- 
mon mode of expressing it, than there is between the Latin word 
equus, a horse, and that most useful of quadrupeds. In each 
case the relations are wholly conventional, and might be altered 
without in any way affecting the object. 



NATURE OF ARITHMETIC. O 

Arithmetic is the science of numbers. Numbers treat of 
magnitude or quantity ; and whatever is capable of increase or 
diminution is a magnitude or quantity. A sum of money, a 
weight, or a surface, is a quantity, being capable of increase or 
diminution. But as we cannot measure or determine any quan- 
tity except by considering some other quantity of the same kind 
as known, and pointing out their mutual relation, the measure- 
ment of quantity or magnitude of all kinds is reduced to this : fix at 
pleasure upon any one known kind of magnitude of the same spe- 
cies as that which has to be determined, and consider it as the 
measure or unit, and determine the proportion of the proposed 
magnitude to this known measure. This proportion is always 
expressed by numbers ; so that number is nothing more than the 
proportion of one magnitude to that of some other magnitude 
arbitrarily assumed as the unit. If, for example, we want. to 
determine the magnitude of a sum of money, we must take some 
piece of known value — such as the pound or shilling — and show 
how many such pieces are contained in the given sum. If we 
wish to express the distance between two cities, we must use 
some such recognized measure of length as the foot or mile ; and 
if we wish to ascertain the magnitude of an estate, we must em- 
ploy some such measure of surface as the square mile or acre. 
The foot-rule is the measure of length most used for engineering 
purposes. The foot is divided into twelve inches, and the inch 
is subdivided into half inches, quarter inches, eighths, and six- 
teenths. It is clear that two half inches or four quarter inches 
make an inch, as also do eight eighths and sixteen sixteenths ; 
and indeed it is obvious that into whatever number of parts the 
inch is divided, we shall equally have the whole inch if we take 
the whole of the parts of it. If the inch were to be divided into 
ten equal parts, then ten of these parts would make an inch. 
Fractional parts of an inch, or of any other quantity, are ex- 
pressed as follows : a half, l ; a quarter, £ ; an eighth, •£ ; a six- 
teenth, T V ; and a tenth, T l ff . The figure above the line is called 
the numerator, because it fixes the number of -halves, quarters, 
or eighths, which is intended to be > expressed; and the figure 
beiOW the line is called the denominator, because it fixes the 



6 ARITHMETIC OF THE STEAM-ENGINE. 

order or denomination of the fraction, whether halves, quarters, 
eighths, or otherwise. Thus in the fractions fths and -|ths, the 
figures 3 and 7 are the numerators, and the figures 4 and 8 the 
denominators ; and fths, fths, or -f^ths, are clearly equal to 1. 
So also fths, fths, and T ^ths are clearly greater than 1, the first 
being equal to l^th, the second to 1-Jth, and the third to l T Vth. 
The species of fractions here referred to is that which is 
ealled vulgar fractions, as being the kind of fractions in common 
use ; and every engineer who speaks of fths or fths of an inch 5 
and every housewife who speaks of f of a pound of sugar, or ^ a 
pound of tea, refers, perhaps unconsciously, to this species of 
numeration. There is another species of fractions, however, 
called decimal fractions, not usually employed for domestic pur- 
poses, but which is specially useful in arithmetical computations, 
and these fractions being dealt with in precisely the same man- 
ner as ordinary figures, are very easy in their application. In 
ordinary figures, the value of each succeeding figure, counting 
from the right to the left, is ten times greater than the preceding 
one, in consequence of its position ; and in decimal fractions the 
value of each succeeding figure, counting from left to right, is 
ten times less. Thus the figures 1111 signify one thousand one 
hundred and eleven ; and if after the last unit we place a period 
or full stop, and write a one after it thus, 1111*1, we have one 
thousand one hundred and eleven and one-tenth. The period, 
or decimal point, as it is termed, prefixed to any number, im- 
plies that it is — not a whole number — but a decimal fraction. 
Thus *1 means one-tenth, # 2 two-tenths, *8 three-tenths, '4 four- 
tenths, and so on. So in like manner *11 means one-tenth and 
one hundredth, or eleven hundredths; '22 means two-tenths and 
two hundredths, or twenty-two hundredths; *83, three-tenths 
and three hundredths, or thirty-three hundredths ; and so on — 
each successive figure of the fraction counting from the left to 
the right, being from its position ten times less than that which 
went before it. The number *1111 signifies one thousand one 
hundred and eleven ten thousandths, the first decimal place 
'being tenths, the next hundredths, the next thousandths, the 
next ten thousandths, and so on. If we wish to express a hun- 



NATURE OF DECIMAL FRACTIONS. 7 

dredth by this notation, we plaoe a cipher before the unit thus, 
•01 ; if a thousandth two ciphers, '001 ; and so of all other quan 
tities. The multiplication, division, and all the other arithmeti 
cal operations required to be performed with decimal fractions, 
are conducted in precisely the same manner as if they were 
ordinary numbers — the decimal progression being carried down 
wards in the one case precisely in the same manner as it is car 
ried upwards in the other case ; and it is easy to suppose that 
the stones used by the natives of Madagascar may not only be 
collected into groups of tens and hundreds, but that each stone 
may also be subdivided into tenths, hundredths, or thousandths, 
so that parts of a stone may be reckoned. Instead of dividing 
the stone into halves, and quarters, and eighths, and sixteenths, 
as would be done by the method of vulgar fractions, it is sup- 
posed by the decimal system of fractions to be at once divided 
into tenths, whereby the same system of grouping by tens, which 
is used above unity, is also rendered applicable to the fractional 
parts below unity — to the great simplification of arithmetical 
processes. In all cases a decimax fraction may be transformed 
into a vulgar fraction of equal value by retaining the significant 
figures as the numerator, and by using as the denominator 1, 
with as many ciphers as there are figures after the decimal point. 
Thus '1 is equal to y 1 ^ ; *11 is equal to T y^ ; *01 is equal to T l$ ; 
•001 is equal to yoVol 3 "145 9 is equal to 3 yWwl an ^ 'T854 is 

equal to yV#o. 

In all countries there are certain recognised standards of 
magnitude for measuring other magnitudes by ; such as the inch, 
foot, yard, or mile for measuring lengths; the square inch, 
square yard, or square mile, or square pole, rood, or acre, for 
measuring surfaces ; the grain, ounce, pound, or ton for measur- 
ing weights ; and the penny, shilling, and sovereign for measur- 
ing money. It is, of course, quite inadmissible in conducting 
any of the operations of arithmetic to confound these different 
lands of magnitudes together, and there is as much difference 
between a linear foot and a square foot as there is between a 
ton weight and a pound sterling. A square surface measuring 
an inch long and an inch broad is a square inch. A strip of sur- 



8 ARITHMETIC OF THE STEAM-ENGINE. 

face 1 inch, broad and 12 inches or 1 foot long will be equal tc 
12 sqnare inches ; and 12 such strips laid side by side, and there- 
fore a foot long and a foot broad, will make 12 times 12 square 
inches, or 144 square inches. In each square foot, therefore, 
there are 144 square inches ; and as there are 3 linear feet in a 
linear yard, there "will be in a square yard 9 square feet, as we 
may suppose the square yard to be composed of three strips of 
surface, each. 3 feet long and 1 foot wide, and therefore contain- 
ing 3 square feet in each. 

A cubic inch is a cube or dice measuring 1 inch long, 1 inch 
broad, and 1 inch deep. A square foot of board 1 inch thick will 
consequently make 144 cubic inches or dice if cut up. But as it 
will take twelve such boards placed upon one another to make 
a foot in depth, or, in other words, to make a cubic foot, it 
follows that there will be 12 times 144, or, in all, 1,628 cubic 
inches in the cubic foot. So, in like manner, as there are 3 lin- 
ear feet in the linear yard, and 9 square feet in the square yard, 
there will be 3 times 9 or 27 cubic feet in the cubic yard — the 
cubic yard being composed of three strata 1 foot thick, contain- 
ing 9 cubic feet in each. 

Besides the square inch there is the circular inch by which 
surfaces are sometimes measured. The circular inch is a circle 
1 inch in diameter, and as it is a fundamental rule in geometry 
that the area of different circles is proportional to the squares 
of their respective diameters, the area of any piston or safety- 
valve or other circular orifice will be at once found in circular 
inches by squaring its diameter, as it is called; or, in other 
words, by multiplying the diameter of such piston or orifice ex- 
pressed in inches by itself. Thus as a square foot, or a square 
of 12 inches each way, contains 144 square inches, so a circular 
foot, or a circle of 12 inches diameter, contains 144 circular 
inches. There is a constant ratio subsisting between a circular 
inch or foot and the square circumscribed around it. The cir- 
cular inch or foot is less than the square inch or foot by a cer- 
tain uniform quantity ; and this relation being invariable, it be- 
comes easy when we know the area of any circle in circular 
inches to tell what the equivalent area will be in square inches, 



SQUARE, CIRCULAR, CUBIC, AND OTHER INCHES. 9 

as we have only to multiply by a certain number — which will 
be less than unity — in order to give the equivalent area. This 
number will be a little more than f , or it will be the decimal 
•7854 ; and if circular inches be multiplied by this number, we 
shall have the same area expressed in square inches. Multiplying 
any quantity by a number less than unity, it may be here re- 
marked, diminishes the quantity, just as multiplying by a num- 
ber greater than unity increases it. To multiply by -| gives the 
same result as to divide by 2 ; and to multiply by the decimal 
•7854 will have the effect of reducing the number by nearly a 
fourth, as it is necessary should be done in order to convert cir- 
cular into square inches ; for, seeing that the square inches are 
the larger of the two, there must be fewer of them in any given 
area. 

Besides the cubic inch there are the spherical, the cylindri- 
cal, and the conical inch, all having definite relations to one 
another. The spherical inch is a ball an inch in diameter ; the 
cylindrical inch is a cylinder an inch in diameter and an inch 
high ; and the conical inch is a cone whose base is an inch in 
diameter, and which is an inch high. All these quantities are 
convertible into one another — just as the pound sterling is con- 
vertible into shillings or pence, and the ton weight is converti- 
ble into hundred-weights and pounds. 

The foundation of all mathematical science must be laid in a 
complete treatise on the science of numbers, and in an accurate 
examination of the different methods of calculation which are 
possible by their means. Now Arithmetic treats of numbers in 
particular, but the science which treats of numbers in general 
is called Algebra. In algebra numbers are expressed by letters 
of the alphabet, and the advantage of the substitution is that 
we are enabled to pursue our investigations without being em- 
barrassed by the necessity of performing arithmetical operations 
at every step. Thus if a given number be represented by the 
letter a, we know that 2 a will represent twice that number, 
and -| a the half of that number, whatever the value of a may 
be. In like manner if a be taken from <z, there will be nothing 
eft, and this result will equally hold whether a be 5, or 7, oi 
1* 



10 ARITHMETIC OF THE STEAM-ENGINE. 

1,000, or any other number whatever. By the aid of algebra, 
therefore, we are enabled to analyse and determine the abstract 
properties of numbers without embarrassing ourselves with 
arithmetical details, and we are also enabled to resolve many 
questions that by simple arithmetic would either be difficult or 
impossible. 

ADDITION. 

The first process of arithmetic is Addition ; and here the 
first steps are usually made by counting upon the fingers, as an 
aid to the perceptions of the total amount of the quantity that 
has to be expressed. For example, if we hold up 5 fingers of 
the one hand and 3 of the other, and are asked how much 5 
and 3 amount to, we at once see that the number is 8, as we 
either actually or mentally count the other 3 fingers from 5, 
designating them as 6, 7, 8; when, the whole fingers being 
counted, we know that the total number to be reckoned is 8. 
Persons even of considerable arithmetical experience, will often 
find themselves either counting their fingers or pressing them 
down successively on the table, in order to assist their memory 
in performing addition. But the best course is to commit very 
thoroughly to memory an addition table, just as the multiplica- 
tion table is now commonly committed to memory by arithmet- 
ical students — as such a table, if thoroughly mastered, will 
greatly facilitate all subsequent arithmetical processes. A table 
of this kind is here introduced, and it should be gone over again 
and again, until its indications are as familiar to the memory as 
the letters of the alphabet, and until the operation of addition 
can be performed without the necessity of mental effort. The 
sign + placed between the figures of the following table is the 
sign of addition termed plus, and signifies that the numbers are 
to be added together. The table is so plain as scarcely to re- 
quire explanation. The figures in the first column are obtained 
by adding together the figures opposite to them in any of the 
other columns. Thus 4 and 9 make 13, as also do 5 and 8 or 8 
and 7. 



METHOD OF PERFORMING ADDITION. 
ADDITION TABLE. 



11 



2 


1 + 1 




3 


1 + 2 






4 


1 + 3 


2 + 2 




5 


1+4 


2 + 3 






6 


1 + 5 


2 + 4 


3 + 3 




1 

8 
9 


1 + 6 


2 + 5 


3+4 






1 + 7 


2 + 6 


3 + 5 


4 + 4 




1 + 8 


2 + 7 


3 + 6 


4 + 5 




10 
11 
12 


1 + 9 


2 + 8 


3 + 7 


4 + 6 


5 + 5 


2 + 9 


3 + 8 


4 + 7 


5 + 6 




3 + 9 


4 + 8 


5 + 7 


6 + 6 




13 


4 + 9 


5 + 8 


6 + 7 






14 


5 + 9 


6 + 8 


7 + 7 




15 


6 + 9 


7 + 8 






16 


7 + 9 


8 + 8 




17 


8 + 9 






18 9 + 9 





GENERAL EXPLANATION OF THE METHOD OF PERFORMING 

ADDITION. 

Write the numbers to be added under one another in such 
manner that the units of all the subsequent lines of figures shall 
stand vertically under the units of the first line — the tens under 
the tens, the hundreds under the hundreds, and so on. Then 
add together the figures found in the units column. If their 
sum be expressed by a single figure, write the figure under, the 
units column, and commence the same process with the tens 



12 ARITHMETIC OF THE STEAM-ENGINE. 

column. But if the sum of the figures in the units column be 
greater than 9, it must in that case he expressed in more than 
one figure, and in such event write the last figure only under the 
units column, and carry to the column of tens as many units as 
are expressed by the remaining figure or figures. Proceed in 
the same manner with the column of tens, and so with all the 
other columns. TVhen the column of the highest order, which 
is always the first on the left, has been added, including the 
number carried from the column last added up, then if the sum 
be expressed by a single figure, place that figure under the col- 
umn. But if it be expressed in more figures than one, write 
those figures in their proper order, the last under the column 
and the others preceding it. 



Examples. 

Add togetner 1,904, 9,899, 5,467, and 2,708. The numbers 

are to be arranged as follows: 

1904 Here, beginning at the right-hand column, we say 8 

9899 and 7 are 15, and 9 are 24, and 4 are 28. "We write the 

nhr Z 8 under the column of units, and carry the 2 tens to the 
2708 ' J 

next column of tens. Adding up this column, we ha\ e 



19,978 the 2 carried from the last column added to 6, which 
make 8, and 9 are 17. Here we write down the 7 and 
carry the 1 over to the next column. In the third column we 
have 1 carried from the last column added to 7, which makes 8, 
and 4 are 12, and 8 are 20, and 9 are 29. Here we write down 
the 9 and carry the 2 to the next column. In the fourth col- 
umn we have the 2 carried from the last column, which added 
to 2 makes 4, and 5 are 9, and 9 are 18, and 1 are 19, which 
sum of 19 we write at the foot of the column, the 9 under the 
other figures and the 1 preceding it. The total sum of these 
several numbers therefore, when added together, is nineteen 
thousand nine hundred and seventy-eight. 
Add together the following numbers : — 



SE OF 


COMMAS IN 


NOTATION— 


-SUBTRACTION. 


2808 


1467 


2708 


5794 


1407 


5988 


5467 


9969 


9969 


2829 


9899 


1407 


5794 


9694 


1904 


2808 


19,978 


19,978 


19,978 


19,978 



13 



It is usual, for facility of reading the figures, to divide them, 
when they amount to any considerable number, into groups of 
three each, by means of a comma interposed. But the comma 
in no way affects the value of the quantity, but is merely used 
to save the trouble of counting the figures to make sure whether 
it is thousands, hundreds of thousands, or what other order of 
figures is intended to be expressed. Thus with the aid of the 
comma we see at once that the number 19,000 is nineteen thou- 
sand, or that the number 190,000 is one hundred and ninety 
thousand, or that the number 1,900,000 is one million nine hun- 
dred thousand ; whereas, without the aid of the commas, we 
should have to count the figures to make sure of the real value 
of the expression. The comma, therefore, has no such signifi- 
cance as the decimal point, and the number may be written 
with or without the comma at pleasure ; but if written without 
it there will be more difficulty in reading the number, just as it 
vfould be more difficult to read a book if the stops were left out. 

SUBTRACTION. 

Subtraction is the reverse of addition. If we have a bag 
containing 20 shillings, and if we add thereto 5 shillings, 15 
shillings, and 10 shillings, we can easily tell by the operation of 
addition that we must have 50 shillings in the bag. If, how- 
ever, we now withdraw the 5 shillings, the 15 shillings, and the 
10 shillings, or, in all, if we withdraw 30 shillings, we shall, of 
course, have the original 20 shillings left ; and the operation of 
subtraction is intended to tell us, when we withdraw a less 
number f»>m a greater, how much of the greater number we 
shall have left. As addition is signified by the sign -f or plus. 



14 ARITHMETIC OF THE STEAM-ENGIXE. 

bo subtraction is signified by the sign — or minus; and two 
short parallel lines = are employed as a substitute for the words 
equal to. As the expression, therefore, 5 + 3 means 5 increased 
by 3, or 8 ; so the expression 5—3 means 5 diminished by 3, oi 
2. This in common arithmetical notation would be written 
5 + 3 = 8 and 5 — 3 = 2. 

When we have a number of quantities to subtract from a 
greater quantity, the usual course is to add together first all the 
quantities to be subtracted, in order that the subtraction may 
be performed at a single operation. Thus in the case of the bag 
containing 50 s hillin gs, from which we successively withdraw 
5 shillings, 15 shillings, and 10 shillings, we first add together 
the 5 shillings, the 15 shillings, and the 10 shillings, so as to 
have in one sum the whole quantity to be subtracted, and then 
we can suppose the operation to be performed at a single step, 
as, the subtraction having been performed at different times, 
will not affect the amount of the sum subtracted or the sum 
left. Thus 50 — 30 = 20; or if we take the successive stages, 
we have 50 — 5 = 45, and 45 — 15 = 30, and 30 — 10 = 20, 
which is the same result as before. 



GENERAL EXPLANATION OF THE METHOD OE PERFORMING STJB- 

TEACTION. 

"Write the less number under the greater in such manner 
that the units of the second line of figures shall stand vertically 
under the units of the first line — the tens under the tens, the 
hundreds under the hundreds, and so on, as in addition. Draw 
a straight line beneath the lower line of figures, and subtract 
the units of the lower line of figures from the units of the up- 
per line, and place the remainder vertically under the units col- 
umn and beneath the straight line which has been drawn. Sub- 
tract the tens from the tens in lite manner, the hundreds from 
the hundreds, and so on until the whole is completed; and 
where there is no figure to be subtracted, the figure of the up< 
per line will appear in the answer without diminution, as ap- 
pears in following examples : 



METHOD OF PERFORMING SUBTRACTION. 15 

1864 Original number 1864 Original number 

64 Number to be subtracted 32 Number to be subtracted 

1800 Remainder 1832 Remainder 



From 7854 From 89764384 From '785068473894 

Subtract 6532 Subtract 41341073 Subtract 510054103784 



Answer 1322 Answer 48423311 Answer 275014370110 



In these examples all the figures of the second line are less 
than those of the first line, and we at once see what the re- 
mainder at each step will be by considering what sum we must 
add to the less number to make it equal to the greater. Thus 
in subtracting 6532 from 7854, we see that we must add 2 to 
the 2 of the lower line to make the 4 appearing in the upper; 
and we must add 2 to the 3 appearing in the lower line to make 
the 5 appearing in the upper. In cases, however, in which 
some of the figures of the lower line are larger than those ex- 
isting in the upper, we must borrow a unit from the preceding 
column, which will count as ten in the column into which it is 
imported, and this unit so borrowed will be added to the sum 
to be subtracted when that preceding column comes to he dealt 
with. Thus in the groups of stones used by the natives of Mad- 
agascar — if we have 6 groups of 10 stones in each and 7 stones 
over, and if we want to withdraw 8 stones from the number, it 
is clear that, as the 7 stones not arranged in groups will not 
suffice to supply the 8 stones we have to furnish, we must break 
up one of the groups of 10 to enable the 8 stones to be surren- 
dered. "We shall then have only 5 groups, hut with the 7 stones 
we had before we can supply the 8 by taking only one stone 
from one of the groups, leaving 9 stones in it, so that, after tak- 
ing away the 8 stones, we shall have 5 groups of ten each and 
9 stones left. This is expressed arithmetically as follows : 

67 Here we say we cannot subtract 8 from 7, so that we 

8 must borrow 1 from the previous column, which, when 

59 imported into the column of units, will be 10 ; and we 

■= therefore say 8 taken from 17 leaves 9, which 9 we place 



16 



ARITHMETIC OF THE STEAM-ENGINE. 



in the remainder. But as we have taken one of the groups 
from the preceding column, we have to deduct that from the 
six groups remaining, and we therefore say 1 from 6 leaves 5. 
So, in like manner, if we had to take 29 shillings from 42 shil- 
lings, as we cannot take 9 from 2, we take 9 from 12, borrow- 
ing as before a unit from the preceding column. But as we 
have afterwards to return this unit, we do not say 2 from 4, but 
3 from 4, which leaves 1 ; or, in other words, 29 taken from 42 
leaves 13, as we can easily see must be the case, as 13 added to 
29 make 42. To prove the accuracy of an answer in subtrac- 
tion, it is only necessary to add together the two lower lines, 
which will produce the top one. 



Examples. 





1864 


Subtract. . . . 


14 


Eemainder 


1850 




1864 


Subtract . . . 


975 


Eemainder 


889 



From 1864 

Subtract ... 97 



Eemainder 1767 



From 1864 

Subtract 1796 



Eemainder 



68 



It will be seen that, by adding together the last two lines of 
figures in each of these examples, we obtain the first line. 



MULTIPLICATION. 

Multiplication is a process of arithmetic for obtaining the 
sum total of a quantity that is repeated any given number of 
times, and is virtually an abbreviated species of addition. If, 
for example, we have 6 heaps of stones, with 1,728 stones in 
each heap, we might ascertain the total number of stones in the 
six heaps by writing the 1,728 six times in successive lines, and 
adding up the sum by the method of procedure already ex- 
plained under the head of Addition. But it is clear that this 
would be a very tedious process in eases in which the number 



MULTIPLICATION A SPECIES OF ADDITION. 17 

of heaps was great, and multiplication, is an expedient for ascer- 
taining the total quantity by a much less elaborate method of 
procedure. 

All numbers whatever it is clear may be formed by the addi- 
tion of units. The consecutive numbers 1, 2, 3, 4, 5, &c, may 
be derived as follows • 

1=1 

1 + 1=2 
1+1+1=3 
1+1+1+1=4 
1+1+1+1+1=5 

There are certain numbers which are formed by the contin- 
ued addition of other numbers than 1 ; and the numbers which 
are formed by the continued addition of 2 may be shown as fol- 
lows: 

2=2 
2 + 2=4 

2 + 2 + 2=6 

2+2+2+2=8 

2 + 2 + 2 + 2 + 2=10. 

In like manner, the numbers shown by the "successive addi- 
tions of 3 and 4 may be thus represented : — 

3=3 4=4 

3 + 3=6 4 + 4=8 

3+3+3=9 4+4+4=12 

3 + 3 + 3 + 3=12 4 + 4 + 4 + 4=16 

3 + 3 + 3 + 3 + 3=15 _ 4 + 4 + 4 + 4 + 4=20 

Thus it will be seen that in the series of numbers proceeding 
upwards from 1, some can only be formed by the continued ad- 
dition of 1, while others may be formed by the continued addi- 
tion of 2, 3, or some higher number. The numbers 3, 5, and 7 
cannot be produced by the continued addition of any other 
number than 1, while the intermediate numbers 4 and 6 may be 
formed, the first by the addition of 2, and the second by the con- 
tinued addition of 2 or 3. 



18 AEITHMETIC OF THE STEAM-ENGINE. 

Those numbers which cannot be formed by the continued 
addition of any other number than 1 are termed prime numbers. 
The numbers 3, 5, 7, 11, 13, 17, &c, are prime numbers. All 
other numbers are termed multiple numbers ; and they are said 
to be multiples of those lesser numbers by the continued addi- 
tion of which they may be formed. Thus 6 is a multiple of 2, 
because it may be formed by the continued addition of 2. But 
it is also a multiple of 3, because it may be formed by the con- 
tinued addition of 3. In like manner 12 is a multiple of 2, 3, 4, 
and 6. 

In the ascending series of numbers, 1, 2, 3, 4, 5, &c, it will 
be obvious that each alternate number is a multiple of 2. Such 
numbers are called even numbers, and the intermediate numbers 
are called odd numbers. Thus 2, 4, 6, 8, 10, &c, are even num- 
bers, and 1, 3, 5, 7, 9, &c, are odd numbers. 

As every even number is a multiple of 2, it is clear that no 
even number except 2 itself can be a prime number, and every 
prime number except 2 itself must be an odd number. It by no 
means follows, however, that every odd number must be prime, 
and it is clear indeed that 9 is a multiple of 3, 15 of 3 and of 5, 
and so of other odd numbers, which cannot, therefore, be prime 
numbers. 

If we take a strip of paper an inch broad and 12 inches long, 
like a strip of postage stamps, it is clear that this strip will con- 
tain 12 square inches ; and if we take three such strips placed 
side by side, they will manifestly have a collective surface of 36 
square inches. Nor will the result be different in whatever way 
we reckon the squares ; and 12 multiplied by 3 will give just the 
same number as 3 multiplied by 12. In like manner, 7 multi- 
plied by 5 is the same as 5 multiplied by 7, and so of all other 
numbers. 

In order to be able to perform the operations of multiplication 
with ease and expedition, it is necessary to commit to memory 
the product of the multiplications of numbers from 1 to 9 ; and 
to enable this to be conveniently done, a table of these primarj 
multiplications, called the Multiplication Table, forms part ot 
the course of arithmetical instruction given at schools, where, 



THE MULTIPLICATION TABLE. 



19 



However, the tables used commonly carry the multiplications up 
to 12 times 12. A table containing all the multiplications neces- 
sary to be remembered is given below ; and it is very material to 
the subsequent ease of all arithmetical processes, that this table 
should be thoroughly learned by heart, so as to obviate the hesi- 
tation and inaccuracy that must otherwise ensue. 

MULTIPLICATION TABLE. 





2 


3 


4 


5 


6 


7 


8 


9 


9 


18 


27 


3G 


45 


54 


63 


72 


81 


8 


16 


24 


32 


40 


48 


56 


64 




7 


14 


21 


28 


35 


42 


49 






6 


12 


18 


24 


30 


36 






5 


10 


15 


20 


25 






4 


8 


12 


16 






3 


6 


9 






2 


4 







To find the product of two numbers by this table, we must 
look for the greater number in the first upright column on the 
left, and for the lesser number in the highest cross row. The 
product of the two numbers will be found in the same cross row 
with the greater number, and in the same upright column with 
the lesser number. Thus 6 times 3 are 18, 6 times 4 are 24, and 
5 times 4 are 20. If we find the number 6 in the first column 
and pass our finger along the same line until we come vertically 
under the 3 in the top line, we find the number 18, which is the 
product required. By the same process we find the numbers 24 
and 20. 

Having committed the multiplication table to memory, we 
are in a condition for performing any multiplication of common 



20 ARITHMETIC OF THE STEAM-ENGINE. 

numbers without difficulty. If, for example, we wish to multiply 
1,728 by 2, we write the 2 under the 8 and draw a line thus : — 

1*728 ^ e ^ en sa y * w ^ ce 8 are 16. "We write down the 6 

2 and carry the 1, which "belongs to the order of tens next 
3456 above, to that order. Twice 2 are 4, and the 1 carried 
■ from the 16 of the last multiplication make 5. The num- 

ber 5 being less than 10, there is no figure to carry in this case. 
We therefore say twice 7 are 14, where again we write 4 and 
carry 1, and twice 1 are 2, and 1 carried over from the last mul- 
tiplication make 3. 

It is clear that the number 1,728 is made up of the numbers 
1,000, 700, 20, and 8, and the result of the multiplication would 
not be altered if we were to multiply these quantities separately 
and add them together. A Saint Andrew's cross or x is the 
pign of multiplication ; and 

1000x2=2000 

700 x 2=1400 

20x2= 40 

8x2= U 



3456 



Here, then, we see we have precisely the same result as in the 
former case. But the first expedient is the simpler, and is there- 
fore commonly used. "We shall also obtain the same result by 
adding 1,728 to 1,728, thus :— 

,^ 9g In this particular case it is as easy to add the number 
1728 to itself as to multiply by 2. But when the multiplica- 
3456 tion proceeds to 6, 8, or any greater number of times, it 
• would be very inconvenient to have to add the number 

to itself 6 or 8 times, and it is much easier to proceed by the 
common method of multiplication here explained. The number 
we multiply w T ith is called the multiplier, and the number we 
multiply is called the multiplicand, while the number resulting 
from the multiplication is called the product. In the above ex- 
ample 2 is the multiplier, 1,728 the multiplicand, and 3,456 the 
product 



MULTIPLIERS CONTAINING CIPHERS. 21 

If tlie multiplier consists of two figures instead of one, tiro 
game mode of procedure is pursued, except that the whole of the 
figures resulting from the multiplication of the higher of the two 
figures is shifted one place to the left. Thus, if the number 
1,728 has to be multiplied by 22, the mode of procedure is as 
follows :— 

Here the arithmetical process of multiplication is 
22 precisely the same with each of the two figures, only 

that in the case of the second multiplication the result- 

_^:^ ing number is set one place more to the left ; and the 
two lines of partial products are then added together 



88,016 f or the answer. It is, therefore, a rule in all multipli- 
cations where the multiplier consists of more figures 
than one, that the first figure of the product shall be set under 
that particular figure of the multiplier with which that particular 
line of multiplication is performed. If instead of 22 the multi- 
plier had been 222, then the operation would have been as 
follows : — 

Here, it will be observed, the same partial product 
2^2 is repeated in every case, but set one place more to the 

. left ; and the several lines of partial products are then 

^■-fi 6 added up for the total product of the multiplication. 
3455 In cases where one of the figures of the multiplier 

is a cipher, the only effect is to shift the figures over to 



" ' ° the left one place, and which may be done by adding a 
cipher to the product if the cipher forms the last figure 
of the multiplier. Thus, 1,728 multiplied by 20, is 34,560, mul- 
tiplied by 200 is 345,600, and multiplied by 2,000 is 3,456,000. 
If the cipher comes in the middle of the multiplier, as in multi- 
plying by 202, we proceed as follows : — 

Here we pass over the cipher altogether, except that 
202 we begin the succeeding line of multiplication one place 

— — more to the left than we should have done if the cipher 

8456 had not been present ; or, in other words, we begin the 

line pertaining to the next figure of the multiplier un- 

' der that figure, just as would be done if any other 

' figure than a cipher intervened. Indeed we might 



22 ARITHMETIC OF THE STEAM-ENGINE. 

"write a line of ciphers as resulting from multiplication by a 
cipher; but as this line could not affect the value of 'the sum 
total, it is left out altogether. In multiplying numbers termi- 
nating with ciphers, Or in multiplying with numbers terminating 
with ciphers, the mode of procedure is to perform the multipli- 
cation as if there were no ciphers, and then to annex as many 
ciphers to the product as there are ciphers in the multiplier and 
multiplicand together. Thus 65,000 multiplied by 3,300 is 
treated as if 65 had to be multiplied by 33, and then five ciphers 
are added to the product to give the correct answer. 

GEXEEAL EXPLANATION OF THE METHOD OF PEEFOEMIXG- 
MULTIPLICATION. 

The foregoing explanations of the method of performing the 
multiplication of numbers will probably suffice to enable all or- 
dinary questions in multiplication to be readily performed. But 
for the sake of clearness, it may be useful to recapitulate the 
several steps of the process. 

Place the multiplier under the multiplicand, as in addition. 
Multiply the multiplicand separately by each significant figure 
of the multiplier, by which we shall obtain as many partial prod- 
ucts as there are significant figures in the multiplier. TTrite 
these products under one another, so that the last figure of each 
shall be under that figure of the multiplier by which it has been 
produced. Add the partial products thus obtained, and their 
sum will be the total product. 

It will often facilitate arithmetical calculations if we have 
committed to memory the products of numbers larger than those 
found in the common multiplication tables, and it is very impor- 
tant that these elementary multiples should be accurately and 
promptly recollected. In the following table the products cf 
numbers are given as high as 20 times 20 : 



MULTIPLICATION TAELE EXTENDING TO 20 TIMES 20. 23 



o 

CM 


o 

o 










OS 
rH 

00 

I— 1 


o 

00 
CO 

o 

CO 
CO 


1—1 

co 

CO 

cq 

CO 


CM 

CO 






.b- 

T— 1 


o 

CO 


CO 

CO 


CO 

o 

CO 


CS 

00 
CM 




CO 


o 

Cq 

CO 


HH 
O 
CO 


00 

oo 

CM 


CM 
b— 
CM 


CO 
IO 
CM 


1 




IO 
rH 


o 
o 

CO 


io 

00 

cq 


o 
.b- 

CM 


CM 


o 

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24 AEITHilETIC OF THE STEAil-EXGDsE. 

DIVISION. 

"When a number has to be separated into two, three, or any 
other number of equal parts, it is done by means of Division, 
■vrhich enables us to determine the magnitude of one of those 
parts, If, for example, we wish to divide 12 inches into four 
equal parts, the length of each of those parts will be 3 inches. 
If we wish to divide it into three equal parts, the length of each 
of the parts will be 4 inches ; or if we wish to divide it into two 
equal parts, the length of each part will be 6 inches. 

The number which is to be decomposed or divided is called 
the dividend-, the number of equal parts into which the number 
sought to be divided is called the divisor, and the magnitude of 
one of those parts obtained from the division is called the quo 
tient. Thus in dividing 12 by 3, 

12 is the dividend, 

3 is the divisor, 

4 is the quotient. 

It follows from this explanation of the process of division, 
that if we divide a number into two equal parts, one of those 
parts taken twice will reproduce the original number ; or if we 
divide it into three equal parts, one of those parts taken three 
times will reproduce the original number. In ail cases, indeed, 
the quotient multiplied by the divisor will produce the dividend. 
Hence division is said to be a rule which teaches us to find a 
number which, multiplied by the divisor, will reproduce the 
dividend. For example, if 35 has to be divided by 5, we seek 
for a number which, multiplied by 5, will produce 35. This 
number is 7, since 5 times 7 is 35. The manner of expression 
employed in this division is 5 in 35 goes 7 times, and 5 times 7 
makes 35. The dividend, therefore, may be considered as a prod- 
uct, of which one of the factors is the divisor and the other the 
quotient. Thus, supposing we have 63 to divide by 7, we en- 
deavour to find such a product that, taking 7 for one of its fac- 
tors, the other factor multiplied by this shall produce exactly 
63. Xow 7x9 is such a product, and, conseauently, 9 is the 
quotient obtained when ^e divide 63 by 7. 



NATURE OF DIVISION. 25 

In the same sense in which multiplication ahove unity may 
be looked upon as a continued addition, so may division be looked 
upon as a continued subtraction. Thus as 7x9 = 7+7+7 + 7 + 
7+7+7+7+7, so also 63-^-9 = 63-7-7-7-7-7-7-7-7. 
This may easily be seen by performing the operation of addition 
or subtraction. Thus 7 and 7 are 14, and 14 and 7 are 21, and 21 
and 7 are 28, and 28 and 7 are 35, and 35 and 7 are 42, and 42 
and 7 are 49, and 49 and 7 are 56, and 56 and 7 are 63. So in 
like manner 63 less 7 are 56, and 56 less 7 are 49, and 49 less 7 
are 42, and 42 less 7 are 35 and 35 less 7 are 28, and 28 less 7 
are 21, and 21 less 7 are 14, and 14 less 7 are 7, and 7 less 7 
is 0. 

"We have seen that when we divide 12 inches by 4, we ob 
tain 3 inches as the quotient. But if we divide 13 inches by 4 
we shall have 4 parts of 3 inches each and 1 inch over, and if 
this inch be also divided into 4 equal parts, each of these parts 
will be one quarter of an inch. Hence 13 inches divided by 4 
gives 3J inches. So if we divide 63 feet into lengths of 7 feet 
each we shall have exactly 9 of such lengths. But if we divide 
64 feet into lengths of 7 feet each, we shall, after having per- 
formed the division, have 1 foot over. This foot is obviously 
just one sixty- third of the total length ; and if we wish to dis- 
tribute this residual foot equally among the whole of the other 
divisions, we must either divide it into 9 equal parts, and add 1 
of these parts to each division, or we must divide it into 63 equal 
parts, and add 1 of these parts to each foot, or 7 of them to each 
division. It follows that 64 divided by 9 is equal to 7^-, or to 
7/3, which is the same thing. So in dividing a plank 50 feet 
long into lengths of 4 feet each, we shall have 12 such lengths in 
the length of the plank, and we shall have 2 feet over. If we 
wish to distribute these 2 feet equally among the 12 divisions, so 
that no part of the plank may be cut to waste, then we must in- 
crease the length of each foot one forty-eighth part of 2 feet, or 
rve must increase the length of each division one-twelfth part 
af 2 feet, or two-twelfth parts of 1 foot. ~Now, as the foot con- 
sists of 12 inches, two-twelfth parts are equal to 2 inches. More- 
■^er. as a twenty-fourth part of a foot is equal to half an inch, 
2 



26 ARITHMETIC OF THE STEAM-ENGINE. 

and a forty-eighth part of a foot is equal to a quarter of an inch, 
it follows that 8 forty-eighth parts are equivalent to 8 quarters 
of an inch, or to 2 inches, as "before. Each division of the plank 
of 50 feet, therefore, must he 4 feet 2 inches long, in order that 
it may he cut without waste into 12 equal lengths. 

If we have a number 50 which we wish to divide by another 
number 12, then we write the number as follows: — 

We say the twelves in 50, 4 times and 2 over, which two- 

2)50 twelfths is written as a vulgar fraction, and forms 
— part of the quotient. But if we wish the answer to 
4 A be in decimal fractions, we place a decimal point after 
the 50, and add thereto any number of ciphers, continuing the 
division in precisely the same manner as if the number were not 
a fraction at all. Thus — 
1°)50-00000 Here we say, as before, the twelves in 50, 4 

times and 2 over, which 2 we carry to the next 

416666, &e. succeeding place of figures, and say the twelves 
in 20 once and 8 over, the twelves in 80 6 times and 8 over, the 
twelves in 80 6 times and 8 over, and so on to infinity. We 
thus see not merely that the fraction fWths or |th, called the 
remainder, is left over when we divide 50 by 12, but that this 
fraction may be expressed decimally under the form of the infi- 
nite series of numbers '16666, &c, which numbers, if carried on 
for ever, will be continually coming nearer to the quantity |ih, 
but will never be absolutely equal to it, though sufficiently near 
thereto to answer all the purposes of practical computation. 

A very little consideration will suffice to show us the reason 
of the process in division in which we carry the residual num- 
ber to the next place of figures immediately succeeding. Thus, 
if we have to divide the number 963 by 3, we may, if we please, 
perform the operation by dividing the whole number into 900, 
60, and 3, and dividing them separately. Xow the third of 900 
is obviously 300, the third of 60 is 20, and the third of 3 is 1, so 
that the third of the total number of 963 is 321. If, however, 
the number which we had to divide by 3 was 954, then in divid- 
ing the constituent numbers as before, we should have the third 
of 900 which is 300, the third of 50 which is 16, leaving 2 over. 



LONG AND SHORT DIVISION. 27 

Tvhich 2 has to be added to the 4 not yet divided, making it up 

to 6 ; and the third of 6 is 2. These numbers may be written as 
follows : — 

900 divided by 3=300 900 divided by 3 = 300 

60 divided by 3= 20 50 divided by 3= 16 

3 divided by 3= 1 6 divided by 3= 2 



321 318 



By the ordinary method of division, the quantity would be 
written thus : — 

Divisor 3)963 Dividend Divisor 3)954 Dividend 

321 Quotient. 318 Quotient. 

Here, in the first example, we say the threes in 9, three times, 
which 3 we write under the 9 ; the threes in 6 twice, which 2 
we write under the 6 ; and the threes in 3 once, which 1 we 
write under the 3. In the second example we say, as before, 
the threes in 9 three times ; but the threes in 5 will only go 
once, leaving 2 as a remainder, which 2 when imported into the 
next inferior place of figures, will count ten times greater, or as 
20 ; and we then say the threes in 24 eight times, which 8 we 
write under the 4. It will be recollected that as the second 
place of figures from the right is groups of tens, two of these 
groups when resolved into units must necessarily be 20. 

The method of division here described is that used when 
any number has to be divided by another number consisting of 
only one figure. It is called Short Division. In the case of 
quantities which have to be divided by numbers consisting of 
two or more figures, this method would not be convenient, and 
another method called Long Division is commonly employed. 
If, for example, we had to divide 4967398 by 37, we may, no 
doubt, perform the question by the method of short division. 
But the remainders, when there are several figures in the di« 



28 ARITHMETIC OF THE STEAM-ENGINE. 

visor, become so large and perplexing, that it is much better to 
employ the method of long division, which is as follows : 

Dividend 
Divisor 37)4967398(134254 Quotient 
37 

126 
111 

157 
148 

93 

74 

199 

185 

148 

148 



Here we first find how many times 37 are contained in 40, 
and it is clear it is contained only once. "We write therefore 1 
in the quotient, and multiply the divisor by it, placing the prod- 
uct under the 49, and we subtract the 37 from the 49, which 
shows that there is a remainder of 12. To this remainder we 
next bring down the figure of the original number which imme- 
diately succeeds the 4-9, and which in this case is 6. "We then 
consider how many times 37 are contained in 126, and we find 
that it is three times. We write the 3 in the quotient, and 
♦nultiply the divisor by it, when we find that the product is 111, 
which sum we subtract from the 126, and find that we have a 
remainder of 15. To this 15 we next bring down the figure of 
the original sum succeeding to that which we brought down be- 
fore, and which in this case is 7, and we consider how many 
times 37 will go in 157. We find that it will go four times, and 
we write the 4 in the quotient as before, and proceed to multi- 
ply the divisor by it, and to subtract the product 148 from the 
159, which will leave a remainder of 9. Carrying on this pro- 
cess until we have successively brought down all the figures of 



DIVIDING BY THE FACTORS OF A NUMBER. 29 

the original sum that had to he divided, we find that the quo- 
tient is 134254, which number, if multiplied by 37, will repro- 
duce the 4967398 with which we set out. When after perform- 
ing the division there is found to he a remainder, it may either 
be written as the numerator of a vulgar fraction in the answer, 
the divisor being the denominator, or a decimal point may be 
introduced after the last figure, and any desired number of ci- 
phers may be added thereto, when, by continuing the division, 
the remainder will be obtained in decimal fractions. 

The operation of division is indicated by the sign -5- and as 
12 x 12=144, so 144-^-12=12. 

In cases in which the divisor is composed of two factors, it 
is a common practice, instead of employing the method of long 
division to divide successively by the two factors by the method 
of short division, which is more rapidly done. Thus if a num- 
ber has to be divided by, say 36, the same result will be ob- 
tained if it is divided by 6 and the quotient be then again di- 
vided by 6. Or, if we have to divide by 42, we may divide by 
6 and then by 7 ; if Ave have to divide by 63, we may divide by 
9 and then by 7 ; and so of all other numbers possessing similar 
factors. 

As, by annexing a cipher at the end of any number, we mul- 
tiply its amount by 10, so by abstracting a cipher from the end 
of any number we divide its amount by 10. Thus 2 x 10=20 
and 20x10 = 200. So also 200-4-10 = 20 and 20-^-10 = 2. If, 
therefore, we have a divisor containing a number of ciphers, we 
may leave them out of the account in performing division : but 
in such case we must count off as decimals an equal number of 
figures as we have excluded of ciphers. Thus l728-j-10=172'8 
or 1728-4-100=17-28 or 1728-^1000=1-728. So 444-^20=22-2 
and 999-f-30=33-3 or 999-^300=3-33. 

GENERAL EXPLANATION OF THE METHOD OF PERFORMING 

DIVISION. 

Short Division. — Divide the first figure of the dividend by 
the divisor, and place the quotient under the same figure of the 
dividend. Prefix the remainder to the next figure of tho divi- 



30 ARITHMETIC OF THE STEAM-ENGINE. 

dend and divide the number thus obtained by the divisor. Place 
the quotient under the second figure of the dividend, and prefix 
the remainder to the third figure of the dividend. Divide the 
number thus obtained by the divisor, and proceed as before, 
continuing this process until you arrive at the units place of the 
dividend, when the division will be complete. 

Long Division. — Write the divisor on the left of the divi- 
dend, separated from it by a line. Place another line to the 
right of the dividend after the units place to separate the quo- 
tient from the dividend — the quotient being afterwards written 
on the right of that line. • 

Count off" from the left of the dividend or from its highest 
place as many figures as there are places in the divisor. If the 
number formed by these be less than the divisor, then count off 
one more. Consider these figures as forming one number, and 
find how often the divisor is contained in that number. It will 
always be contained in it less than ten times, and therefore the 
quotient will always consist of a single figure. Place this sin- 
gle figure as the first figure of the quotient. 

Multiply the divisor by this single figure, and place the prod- 
uct under those figures of the dividend which were taken off on 
the left, and subtract such product from the number above it, 
by which we obtain the first remainder. This remainder must 
always be less than the divisor. 

On the right of the first remainder place that figure of the 
dividend which next succeeds those which were cut off to the left. 
Find how often the divisor is contained in the number thus 
formed, and place the resulting figure of the quotient next to 
the figure of the quotient already found. Multiply the divisor 
by this figure, and proceed as before, until all the succeeding 
figures of the dividend have been brought down, when the di- 
vision will be complete. 

NATUEE AXD PEOPEETIES OE EEACTIOXS. 

It has already been stated that a fraction which has the same 
numerator and denominator is exactly equal to 1, and therefore 



NATURE AND PROPERTIES OP FRACTIONS. 31 

such a fraction is of the same value as an integer or whole num- 
ber. For example, the fractions 

£ 3 A 5 jS -7. -8. 1 A™ 
23 o3 -±3 53 til 11 83 9? °^«i 

are all equal to 1, and are therefore equal to one another. 

All fractions of which the numerator is less than the denom- 
inator have a less value than unity ; for if a number be divided 
by another number greater than itself, the result must be less 
than 1. If we cut a lath 2 feet long into three equal lengths, 
one of those lengths will certainly be shorter than a foot. 
Hence it is evident that f is less than 1, and for the reason that 
the numerator 2 is less than the denominator 3. If, on the con- 
trary, the numerator be greater than the denominator, then it 
will follow that the fraction will be greater than 1. Thus § is 
greater than 1, for f- is equal to § and |-, and as § is equal to 1, 
then § will be equal to 1|. In the same manner f is equal to 1-|-, 
§ to If, |- to 2|-, and so on. In all such cases it is sufficient to 
divide the upper number by the lower, and if there is a remain- 
der, to write it as the numerator of the residual fraction, and 
the divisor as the denominator. If, for example, the fraction 
were ff, we should divide the 43 by 12, when we should get 3 
as the integer and T 7 ¥ as a remainder ; or, in other words, we 
should obtain the number 3 T 7 ^-. Fractions like -ff, which have 
the numerator greater than the denominator, are termed im- 
proper fractions, to distinguish them from fractions properly so 
called, which, having the numerator less than the denominator, 
are less than unity, or an integer. 

As we can only understand what the fraction T 7 ^ is when we 
know the meaning of -j 1 ^, we may consider the fractions whose 
numerator is unity as the foundation of all others. Such are 
the fractions 

ll-l l l 'l li l l i_ l fop 

'HI ¥3 4? 5) "6"? 73 81 93 103 113 123 13? °^'3 ■ 

and it is observable that these fractions go on continually dimin- 
ishing, for the more we divide an integer, or the greater the 
number of parts into which we distribute it, the less does each 
part become. Thus yi-g- is less than -^ ; t -^q-o i s ^ ess than -^-^ ; 
iuwo i s l ess ^ an toVo \ an( l as we increase the denominator of 



32 ARITHMETIC OP THE STEAM-ENGINE. 

the fraction, the less does the value of the fraction become. If, 
therefore, we suppose the denominator to bo made infinitely 
large, the fraction will become equal to nothing. To express 
the idea of infinity, we make use of the symbol go, and we may, 
therefore, say that the fraction ^r=0. Now, we know that if 

we divide the dividend 1 by the quotient -55-, which is equal to 
nothing, we obtain again the divisor 00 . Hence we learn that 
infinity is the quotient arising from the division of 1 by 0., Thus 
1 divided by expresses a number infinitely great. But £ is 
certainly only the half of f , or the third of § ; so that it would 
appear as if one infinity may be twice or three times greater 
than another. It will be obvious that as the fractions 

23.4_5_67.£9. SL. n 
2"5 35 47 5? 65 75 85 8? <*'<-'• J 

are all equal to one another, each of them being in fact equal to 

1, so also the fractions 

2 4. .8 .10 12 14_ fan 
T5 25 "35 45 5 5 6 5 7 5 ^^«) 

are all equal to one another, each of them being in fact equal to 
2 ; for the numerator of each divided by the denominator gives 

2. So likewise the fractions 

36 912 1518.21 fan 
T5 "25 35 4 5 ~T~1 ~6~5 7 5 °^'l 

are equal to one another, since in fact each of them is equal to 3. 
£Tow it is clear that as £ is the same as ^ and as f is the 
same as - 1 /, both being equal to 3, the value of a fraction will 
not be changed if we multiply numerator and denominator by 
the same number. Thus in the case of the fraction £, if we 
multiply numerator and denominator by 4, we shall have £ 
which is clearly equal to -J. So also the fractions 

12 3 4 5 G 7 _§_ 10 fan 
25 45 65 ¥5 T05 1^5 145 165 TGI u '' t "> 

tiro equal, each of them being equal to -§•. The fractions 

12 3 4 _5_ _6_ _7 8_ 9_ 10 fan 
35 "65 95 l~Zl 155 18) 2 D "245 2 75 305 ""^'j 

ure also equal, each being equal to I ; and the fractions 

• 2. 4 6 _8_ 10 12 14 1G 18 fan 

85 C5 "95 125 "155 lfc» 2 15 ^45 2 75 "^'J 

are also equal, each of them being equal to ^. 



REDUCING FRACTIONS TO LOWEST TERMS. 33 

"Now of all the equivalent fractions 

f46 8 13 14 16 18 <fy-p 
> ¥) "5"? IS"? TB~j #Tj ¥¥? FT) ° iU ? 

the quantity f is that of which it is easiest to form a definite 
idea. It is usual, therefore, when we have any such fraction as 
hi Gr hfi ^° reduce it to its lowest ter?ns, by dividing numerator 
and denominator by some number that will divide each without 
a remainder. This division, it is clear, will not affect the nu- 
merical value of the fraction ; for if we can multiply both nu- 
merator and denominator by the same number without affecting 
the value, so we may divide both without affecting the value ; 
as by such division we bring back the fraction of which both 
portions had been multiplied to the original expression. 

The number by which the numerator and denominator of a 
fraction may be divided without leaving a remainder is called a 
common divisor ; and so long as we can find for the numerator 
and denominator a common divisor, it is certain that the frac- 
tion may be reduced to a lower form. But if we cannot find 
such common divisor, the fraction is in its lowest form already. 
Thus in the fraction -j 4 ^-, we at once see that both terms are di- 
visible by 2, and, performing this division, the fraction becomes 
14 ; which, if again divided by 2, becomes Jf , and which in like 
manner, by another division by 2, becomes T 6 T . This, it will be. 
obvious, cannot any longer be divided by 2, but it may be by 3, 
when the expression becomes f ; and as this cannot be divided 
by any other number than 1, it follows that the fraction is now 
in its lowest terms. JSTow 2x2x2x3=24, and instead of the 
successive divisions by 2, by 2, by 2, and by 3, we may divide 
at once by the product of these quantities, or 24; and dividing 
numerator and denominator of -££<$ by 24, we have f as before. 

The property of fractions retaining the same value, whether 
we multiply or divide their numerator and denominator by the 
same number, carries this important consequence — that it ena- 
bles fractions to be easily added or subtracted, after having first 
brought them to the same denomination. If, for example,® we 
had to add together f, £, T V, and o% of an inch, we could not do 
so easily unless we brought the whole of these quantities to 
2* 



34 ARITHMETIC OF THE STEAM-EXGESE. 

thirty-seconds. TVhen so reduced the quantities will be |f . gV, 
■fz, -fz, the snm of "which is clearly -=4. or, dividing numerator 
and denominator by S, the expression becomes f. 

All -whole numbers, it is clear, may be expressed by frac- 
tions, since any whole number may be divided into any number 
of parts. Tor example, 6 is the same as f. It is also the same 
as i.f-. J A -M-, - 3 6 6 -, and an infinite number of other expression; 
which all have the same value. 

ADDITIOX AXD BUBTEACTIOH OF FSACTIOXS. 

TThen fractions have the same denomination there is no 
more difficulty in adding or subtracting them than there is in 
adding or subtracting whole numbers. Thus -£+f- is manifestly 
|, and I — | is obviously f. So afc :< 

7l_ 12 15 1 20 9 

IOU'100 100 100~100 100* 

2 A 7 U? I SI 3 6 rr 18 

50 50 jOTjO si wl WS m 

iS 3 1 1 1 14 16 n r, ± 

2 T^ To^TO ^0 Ui 5* 

Also |+|=|=1 and f-f +i=£=0. 

But when fractions have not the same denominators, then, 
before we can add or subtract them, we must change them for 
others of equal value which have the same denominators. For 
example, if we wish to add the fractions 4- and A, we must con- 
sider that i is the same as f, and that £ is equivalent to §. We 
have, therefore, instead of the fractions first proposed, the equiva- 
lent fractions ■§ and f , the sum of which is f • If the two fractions 
were united by the sign—, we should have i— 1- or f- — §—-|-. 
Again, if the fractions proposed be f +f, then as f is the 
same as f, the sum will be f-ff= J ^=lf. If the sum of 
£ and J were required, then as i=^ and i = - 1 3 o-< tn e sum 

These cases are simple and easily reduced. But we may 
have a great number of fractions to reduce to a common denom- 
ination, which require a more elaborate process. Tor example, 
we $iay have 4-. f, -f. f, f, to reduce to a common denomination, 
in order that we may add them together. The solution of such 
a case depends upon finding a number that shall be divisible bj 



TO REDUCE FRACTIONS TO A COMMON DENOMINATION. 35 

All the denominators of these fractions. Here we proceed ac- 
cording to the following rule : 

TO EEDFCE FRACTIONS TO A COMMON DENOMINATION. 

Eule. — Multiply each numerator into every denominator 
except its own for a new numerator, and multiply all the denom- 
inators together for a common denominator. 

When this operation has been performed, it will be found 
that the numerator and denominator of each fraction have been 
multiplied by the same quantity, and consequently that the frac- 
tions retain the same value, while they are at the same time 
brought to a common denomination. 

Example. Beduce -|, f, •£-, A, and -§, to a common denomina- 
tion. 

1x3x4x5x6=380 and 360-f-6= 60 and 60-^-2=30 
2x2x4x5x 6=480 and 480-^6 = 80 and 80-f-2=40 
3x2x3x5x6=540 and 540-^6= 90 and 90-=-2=45 
4x2x3x4x6=576 and 576-f-6= 96 and 96-^-2=48 
5 x 2 x 3 x 4 x 5=600 and 600-f-6=100 and 100-j-2=50 



2 x 3 x 4 x 5 x 6=720 and 720^-6=120 and 120-^2=60 

Here, then, we first multiply 1, which is the numerator of 
the fraction |-, by the denominators of all the other fractions in 
succession. We next multiply the number 2, which is the nu- 
merator of the fraction f, by the denominators of all the other 
fractions — excepting always its own denominator — and we pro- 
ceed in this manner through all the fractions whatever their 
number may be. "We next multiply all the denominators to- 
gether for the common denominator. Proceeding in this way 
we find the first numerator to be 360, the second 480, the third 
540, the fourth 576, and the fifth 600 ; while the new denomi- 
nator we find to be 720. It is clear, however, that these frac- 
tions are not in their lowest terms, and that the numerator and 
denominator of each may be divided by some common number 
without leaving a remainder. We may try 6 as such a divisor, 
and we shall find that the numerators will then become 60, 80, 
90, 96, and 100, and the denominator 120. These numbers. 



36 ARITHMETIC OF THE STEAM-ENGINE. 

however, are still divisible by 2, and performing tbe division 
the numerators become 30, 40, 45, 48, and 50, and the denomi- 
nator becomes 60. The same result would have been attained 
if we had divided at once by 12. And as we cannot effect any 
further division upon all of the numbers by one common num- 
ber, without leaving a remainder in the case of some of them, 
the fractions, we must conclude, are now in their lowest com- 
mon terms. To add together these fractions we have only to 
add together the numerators, and place the common denomina- 
tor under the sum. Performing this addition we find that in 
this case we have ^^, and as -f-g- are equal to 1, it follows that 
Sgig 3 - are equal to 3 and -§-§, or 3 |-i. 
It is easy to prove that the fractions 

21 3) 4) 5? alJAX ¥ 

are of precisely the same value as the fractions 

3.0. 40 4_5 4 8 JLO 
60? 605 60) 60) 60 

which have been substituted for them. Dividing numerator and 
denominator of the first term by 30 we obtain J ; dividing nu- 
merator and denominator of the second term by 20 we obtain 
| ; 15 is the divisor in the case of the third term when we ob- 
tain f ; 12 is the divisor in the case of the fourth term when 
we obtain the fraction f ; and 10 is the divisor in the last case 
when we obtain the fraction f . Dividing the numerator and 
denominator of each of the transformed fractions, therefore, by 
the greatest number that will divide both without a remainder, 
we' get the fractions 

i.i si 4) o) aua 6" 
which, it will be seen, are the fractions with which we set out, 
and they are now in then* lowest terms, but are no longer of 
one common denomination. The lowest terms with a common 
denominator are 

3 40 4 5 4 8 nTl r] 5° 
3"o) 6"o) "(To) 75"o) " ,UA1 7>o" 

as determined above. 

The subtraction of fractions from one another is accom- 
plished by reducing them to a common denomination as for ad- 



ADDITION AND SUBTRACTION OP FRACTIONS. 37 

dition, and then by subtracting the less numerator from the 
greater. Thus if Ave have to subtract f from f, we must re- 
duce them to a common denomination by the process already 
explained, when the first becomes £f, an ^ the second |-f-> s0 that 
f- exceeds f in magnitude by ^ So also if we have to subtract 
f- from -f, the first fraction becomes by the process of reduction 
f £, and the second ff, so that f taken from f leaves i. 

As whenever the numerator of a fraction is a larger number 
than the denominator, the value of the fraction is greater than 
unity, and is equal to unity when numerator and denominator 
is the same, we have only to divide the numerator by the de- 
nominator to find the number of integers which the fraction 
contains. So in subtracting a fraction from a whole number, 
we must break one or more integers up into fractions of the 
same denomination as that which has to be subtracted. Thus 
if we have to take f ^ from 1, we must instead of the 1 write 
££, and §— taken therefrom obviously leaves -£§. If we have to 
add together such sums as 3|- and 2-|, we see at once that the 
whole numbers when added will be 5, and the equivalent frac- 
tions under a common denominator will be f and f or -|, which 
is 1£, so that the total quantity will be 6^. 

The addition and subtraction of decimal fractions are per- 
formed in precisely the same way as the addition and subtrac- 
tion of whole numbers — the only precaution necessary being to 
place the decimal point in the proper place. Thus 78963-874 + 
83952-2 + 364-003 + 10000-997 are added together as follows: 
'"8963-S > "4 Here, beginning as in the addition of whole 
83952-2 numbers with the first column to the right, we find 
364-003 that 7 and 3 are 10 and 4 are 14. TVe set down 
the 4 beneath the column and carry 1 to the next 



173281-074 column. Adding up the next column, we find only 
~ two significant figures in it, and we say 1 added to 
9 makes 10, which added to 7 makes 17. "We set down the 7 and' 
carry the 1 as before to the next column, which when added up 
we find to be 20. This means 20 tenths, and we set down the 
and carry the 2 to the next column just as in simple addition. 
So likewise in subtraction, if we take 2*25 from 4*75, the result 



38 AKITHMETIC OF THE STEAM-ENGINE. 

will be 2'50; or if we take 1*79 from 3, the result is 1 21. In 
such a case we write the 8 thus : 

3-00 Here we write the 3 with a decimal point after it, 

1*79 and we add as many ciphers after the decimal point as 
, 7 there are decimal figures to he subtracted, or we suppose 
- those ciphers to be added. This does not alter the value 
of the 3, as 3 with no fractions added to it is just 3. Perform- 
ing the subtraction we say 9 from 10 leaves 1, and 8 taken from 
10 leaves 2, and 2 from 3 leaves 1, just as in simple subtraction. 

MULTIPLICATION AND DIVISION OF FEACTIONS. 

If we wish to multiply a fraction any number of times, it is 
clear that it is only the numerator we must multiply. Thus if 
we multiply -J of an inch by 3, it is obvious that we shall get | 
of an inch as the product of the multiplication, or ^ repeated 
3 times. We have already seen that to multiply both terms of 
a fraction by any number does not alter the value of the frac- 
tion, and if we were to multiply the numerator and denomina- 
tor of the fraction ■§■ by 3 we should get ^, which is just the- 
same as -§-. Thus also — 

3 times |- makes § or 1J. 
3 times -J- makes § or 1. 

3 times £ makes f or \. 

4 times T 5 o makes ff- or 1^- or If. 

Instead, however, of multipljdng the numerator, we may 
attain the same end by dividing the denominator, and this is a 
preferable practice when it can be carried out, as it shortens the 
arithmetical operation. Thus i multiplied by 2 is f or \. But 
by dividing the denominator of \ by 2, we obtain the same 
quantity of J at one operation. So also if we have to multiply 
f by 3 we obtain -2gi or f. But if, instead of multiplying the 
numerator, we divide the denominator, we obtain the f at one 
operation. In the same way-|-f multiplied by 6 is equal -^, or 3J. 

"Where the integer with which the multiplication is per- 
formed is exactly equal to the denominator of the fraction, the 
product will be equal to the numerator. Thus — 



MULTIPLICATION OF FRACTIONS BY FRACTIONS. 39 
\ X2 = l 

fx3 = 2 

f x4 = 3 
Having now shown how a fraction may be multiplied by an 
integer, the next step is to show how a fraction may be divided 
by an integer ; and just as a fraction may be multiplied by di- 
viding the denominator, so may a fraction be divided by multi- 
plying the denominator. It is clear that if we divide half an 
inch into two parts, each of these parts will be J of an inch, 
and we divide quarter of an inch into two parts, each of those 
parts will be -|- of an inch, so that J-f-2 =£ and £-s~2 — •§-, which 
quantities we obtain by successively multiplying the denomina- 
tors. We may accomplish the same object by dividing the nu- 
merator where it is divisible without a remainder. Thus f- di- 
vided by 2 is clearly -£, and f divided by 3 is f . Thus also 

J£ divided by 2 gives fa 

H divided by 3 gives ^, 

l~l divided by 4 gives ¥ 3 -. 
When the numerator is not divisible by the divisor without 
a remainder, the fraction may be put into some equivalent form, 
when the division may be effected. Thus if we had to divide 
| by 2, we might turn it into the equivalent fraction f , which, 
divided by 2, gives £ . But the same number is more conven- 
iently found by multiplying the denominator instead of by di- 
viding the numerator. 

We have next to consider the case where one fraction has to 
be multiplied by another. Thus if the fraction § has to be mul- 
tiplied by the fraction -f , we have first to remember that the ex- 
pression f means 2 divided by 3, and we may first multiply by 4, 
which produces f, and then divide by 5, which produces T 8 ^. 
Hence, in multiplying a fraction by a fraction, we multiply the 
numerators together for the new numerator, and the denominat* 
ors together for the new denominator. Thus, 

\ x f gives the product £ or £, 

f-x-f gives T 8 T , 

| x -^ gives H or f 6 . 



40 ARITHMETIC OF THE STEAM-ENGINE. 

Finally, we have to show how one fraction may be divided 
by another. If the two fractions have the same number for a 
denominator, the division takes place only with respect to the 
numerators. An inch being ■£% of a foot, it is clear that -^ is 
contained in -^ just as often as 3 inches is contained in 9 inches 
or 3 times ; and in the same manner, in order to divide ■£% by 
T 9 2 , we have only to divide 8 by 9, which gives f . So also ^ ff is 
•contained 3 times in §•§■, and T ^ 9 times in T 4 r 9 g. But when the 
fractions have not the same denominator, then we must reduce 
them to a common denominator by the method of reduction al- 
ready explained. This result, expressed in words, will be as 
follows : — Multiply the numerator of the dividend by the denom- 
inator of the divisor for the new numerator, and the denomi- 
nator of the dividend by the numerator of the divisor for the new 
denominator. Thus -f divided by f =-£-§- , and f divided by |"=f or 
|, or 1-|, and f-f divided by f ='Jf g- or -§. This rule is commonly 
expressed in the following form : — Invert the terms of the divisor 
so that the denominator may be in the place of the numerator. 
Multiply the fraction which is the dividend by this inverted frac- 
tion, and the product will be the quotient sought. 

Thus f divided by \=% x f =£ =1£. Also, \ divided by f =f x 

3 — 15 nnrl 2 5 rHvi'-lprl hv 5 — 25y6 — 150 nv £- 

If we have a line 100 feet long, and if we divide it in half, we 
shall manifestly have two lines each 50 feet long. So if we di- 
vide it into lengths of 25 feet, we shall have 4 such lengths ; if 
we divide it into lengths of 2 feet each, we shall have 50 such 
lengths ; and if into 1 foot lengths, we shall have 100 of them ; 
if into lengths of half a foot, we shall have 200 lengths; and if 
into lengths of \ of a foot, we shall have 400 such lengths. 
Hence 

100 divided by 100 = 1 

100 divided by 50=2 

100 divided by 

100 divided by 

100 divided by 

100 divided by 

"We see, therefore, that to divide a number by the fraction \ 



25 = 


= 4 


1 


100 


i . 

'2" 


= 200 


l- 


= 400 



DIVISION OF FRACTIONS BY FRACTIONS. 41 

js equivalent to multiplying it by 2 ; to divide by the fraction | 
is the same as to multiply by 4. So, further, if we divide 1 by 
the fraction yoVo- the quotient is 1,000, and 1 divided by l0 ^ 06 
is 10,000. As, then, the fraction gets smaller and smaller, the 
quotient gets greater and greater, so that we are enabled to con- 
ceive that a number divided by will be indefinitely great, since 
iu fact there will be an indefinitely great Dumber of nothings in it. 
As every number whatever, divided by itself, produces unity, 
so a fraction, divided by itself, produces unity. Thus f -s-| = | X 

4 — 1 

The multiplication of decimal fractions is performed in pre- 
cisely the same way as the multiplication of whole numbers, and 
we must mark off in the product as many decimal places as there 
are in the multiplier and multiplicand together. Thus 1*0025 
multiplied by 2*5 = 2-50625 ; also, "0048 multiplied by -000012 
= 0000000576. 

The division of decimals is performed in the same way as the 
division of common numbers; and if the number of decimal 
places in the divisor be the same as in the dividend, the quotient 
thus obtained will be the quotient required, and will be a whole 
number. But if the number of decimals in the dividend exceed 
that in the divisor, mark off in the quotient obtained by this di- 
vision as many decimal places as make up the difference. But 
if the number of decimals in the divisor exceed that in the divi- 
dend, annex 6s many ciphers to the quotient as make up the dif- 
ference. Thus "805 divided by 2'3 = *35, and 2*5 divided by 
•32 = 7-8125. 

The number 3*045 denotes 3 units, tenths, 4 Jiundreths, and 

5 thousandths, and it might be written 3+ T o+- I -jj- s - + T ^ ir , and 
the number 3*47 might be written 3 -j--^-"^-^, or it might bt 

wr.ten 300 + 4Q + ^ == g47 > g a i so 13-76 = l&JWW == 13£, and 

100 100 10 ° ' 

23-0625 = 23^1, = 23 T V Also, 4-35 = 4+ T 8 <> + t£o- or to -f -f 
•&+to"o ; or by reducing the fractions to the same denomination 
it is $%%+ 30 + T ^=*£|. So ff|, put in the form of a decimal, 
wll be 5'62, for m=m+T%°o + t£o- But fg£=l, and there! 
T 6 o°o-=T fi c, and 5+A+t*t=5'62. 



42 ARITHMETIC OF THE STEAM-ENGINE. 

PROPORTION. 

The Proportion or Ratio of one quantity to another is the 
number which expresses what fraction the former is of the lat- 
ter, and is therefore obtained by dividing the former by the 
latter. 

The most distinct idea, of proportion is obtained hj reference 
to a triangle such as that here figured, where ab has the samo 
proportion tc bc that ad has to de. It is clear that if the quan- 
tities ab, ad, and eg are fixed, the quantity de will also be de- 
termined, as we have only to draw the line ae through c until 

Ffe.1. 




it intersects the vertical line de, which it will thereby cut off to 
the proper length. Thus also the ratio 108 to 144, or as it is 
written 108 : 144, is ^ff = f . A proportion is usually stated as 
follows : 2 is to 4 as 4 is to 8, or 2 : 4::4 : 8 ; and in all cases 
of proportion the product of the first and fourth terms are equal 
to the product of the second and third terms. This is expressed 
by saying that the product of the extremes is equal to the prod- 
uct of the means. So 2 x 8 = 4 x 4. Conversely, if the product 
of any two numbers equal the product of other two, then the 
four numbers are proportionals. The method by which we find 
a fourth proportional to three given quantities, by multiplying 
together the second and third and dividing by the first, is what 
is termed the Bexe of Theee. If a yard of calico costs 1 shil 
liug, it is clear that 20 yards will cost 20 shillings; and we say, 
therefore, 1 yard is to 20 yards as 1 shilling is to 20 shillings ; 
or we say, 3 inches: 12 inches :: 12 inches: 48 inches. Here 
we obtain the 48 by multiplying together 12 and 12, which 
makes 144, and which divided by 3 gives 48. 

Proportion is in fact a mere question of scale. If we mate 



NATURE OF PROPORTION. 



43 



a mode] or drawing of a house or a machine, we inay make it on 
the scale of J of an inch to the foot, or ■£ an inch to the foot, or 
1 inch to the foot, or 1-| inches to the foot, or on any scale what- 
ever. But the object, when constructed of the full size, will be 
precisely the same on whatever scale the model or drawing has 
been formed. If the scale be £ of an inch to the foot, then it is 
clear the object when formed of full size will be 48 times larger 
than the model or drawing — that is, it will be 48 times longer, 
£8 times broader, and 48 times higher. So in like manner if the 
h inch scale be employed, the object will be 24 times larger; if 
the scale be 1 inch, it will be 12 times larger ; and if the scale 
be 1£ inches to the foot, it will be 8 times larger. So in like 
manner £20 bears the same proportion to £1 that 20 shillings 
bears to 1 shilling. But £20 are 400 shillings, and £1 are 20 
shillings. Hence, by transforming the pounds into shillings, we 
see that 400 shillings bear the same relation to 20 shillings that 
20 shillings bear to 1 shilling; or, in other words, 400 : 20 : : 
20: 1. 

If we take a rectangular figure such as abcd, say 4 inches 
long and 1 inch wide, and if we enlarge this figure by making it 
4 inches longer and 4 inches broader, we see at a glance that the 
resulting rectangle aefg is not of the same shape, and in fact is 
not the same kind of figure as the original rectangle abcd. This 




.s because the enlargement was not made proportionally, and 
the diagonal af consequently does not lie in the same line as the 



44 



ARITHMETIC OF THE STEAM-ENGINE. 



diagonal ao. To make the enlargement proportional, we should 
only have extended ab 1 inch, when we extended ad 4 inches 



Eiff. 3. 




Such an extension is shown by the rectangle aihg; and the 
diagonal cf that rectangle lies in the same hue as that of the 
original rectangle abcd. In like manner, if the elliptical figure 
ab be enlarged by equal quantities in the line ab and in the lino 
cd, each successive ellipse becomes more circular, and to main- 
tain the original figure the enlargements should be in the pro- 
portion of the length and breadth. 



ON THE SQUAEES AND SQTJAEE BOOTS OF NUMBEES. 

The product of a number multiplied by itself is called a square, 
and the quotient obtained by dividing this product by the num- 
ber is the square root of the product. Thus 12 jimes 12 is 144, 
which is the square of 12 ; and 144 divided by 12 is 12, which is 
the square root of 144. In like manner, the square root of 12 is 
the particular number which, multiplied by itself, produces 12. 
Such number is neither 3 nor 4, as 3 times 3 is 9 and 4 times 4 
is 16, of which the one is less than 12 and the other greater. 
The square root of 12 will be some number between 3 and 4, and 
what the particular number is it is the object of the process for 
determining square roots to discover. The origin of the term is 
traceable to the language cf geometry, where a rectangular sur- 
face is produced by the multiplication of one linear dimension 



SQUARES OF INTEGERS AND FRACTIONS. 45 

with another, or a square is produced by the multiplication of 
one linear dimension by itself. Thus a piece of board a foot long 
and a foot broad has a surface of one square foot, or, if we count 
the dimensions in inches, as the length is 12 inches and the 
breadth 12 inches, the superficies will be 12 times 12, or 144 
square inches. The square of 1 is 1, since 1 x 1=1. The square 
of 2 is 4, since 2 x2=4. The square of 3 is 9, since 3 x3=9. 
Contrariwise 1, 2, and 3 are the square roots of 1, 4, and 9. 
If we write the numbers 

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 
and their squares 

1, 4. 9, 16, 25, 36, 49, 64, 81, 100, 122, 144, 169, 

(t will be seen that if each square number is subtracted from that 
which immediately follows, we obtain the series of odd numbers 

3, 5, 7, 9, 11, 13, 15, 17, 19, 21, &c, 

in which the numbers go on increasing by 2. 

The square of a fraction is obtained by multiplying the frac- 
tion by itself, in the same manner as a whole number. Thus 

2*2 — <£ 5 3 •*• o — 95 8*3 — 91 4*¥ — 165 <*LLU ^ A a — l6 « UU 

also ^ is the square root of £; -J is the square root of £, and i is 
the square root of j 1 ^. 

When the square of a mixed number, consisting of an integer 
and a fraction, has to be determined, we may reduce the mixed 
number to a fraction by multiplying the integer by the denomi- 
nator, and adding the numerator to form a new numerator with 
the same denominator for the denominator of the new fraction. 
Thus 3|=^-+|=-V- and the square of -%'-=-\\ l - or 15 -V Thus 
also, as the square of -f is f f , the square root of f f is -f, and the 
square root of 12^ or 4 ? 9 -=|=3-|. But when the number is not 
a square, it is impossible to extract its square root precisely, 
though the root may be approximated to with any required de* 
gree of nearness. "We have already seen that the square root of 
12 must be more than 3 and less than 4. We have also seen, 
.hat tliis root is less than 3|-, as the square of 3-§- is 12£. Neither 
is the root d T \ or f| the square of which is ^-J- or 12of ? , which 



"46 ARITHMETIC OF THE STEAM-ENGINE. 

is still greater than 12. So if we -try the number 3 T 6 3 or 5 i °-^» 
we shall find the number to be too small, for 12 reduced to the 
same denomination is -yW-j so * na * &t% ^ s two t°° smau > while 
3 T 7 5 is too great. The fact is, whatever fraction we annex to 3, 
the square of that sum will always contain a fraction, and will 
never be exactly 12 ; and although we know that 3 -^ is too great, 
and 3/g- is too small, we cannot fix upon any intermediate num- 
ber which multiplied by itself shall produce 12 ; whence it fol- 
lows that the square root of 12, though a determinate magnitude, 
cannot be expressed by fractions. There is therefore a kind of 
numbers which cannot be specified by fractions, but which still 
are determinate quantities, and of these numbers the square root 
of 12 is an example. These numbers are called irrational num- 
bers, and they occur whenever we attempt to find the square root 
of a number that is not a square. These numbers are also called 
turds or incommensuraoles. The square roots of all numbers 
which are not perfect squares, are indicated by the sign ^/, which 
is read square root. Hence a/12 means the square root of 12; 
^/2 the square root of 2 ; ^/B the square root of 3 ; •y'l the square 
root of § , and *Ja the square root of a. As, moreover, the square 
root of a number multiplied by itself will produce the number, 
^2 multiplied by ^/2 will produce 2 ; ^/3x^/3=3; ^5x^5=5; 

Vt x Vf = t ? anc ^ V a x V a produces a. 

Although these irrational quantities cannot be expressed in 
fractions, it will not therefore be supposed that they are visionary 
or impossible. On the contrary, they are real quantities, which 
may be dealt with in the same way as common numbers ; and 
however difficult of appreciation such a number as the square 
root of 12 may be, we at least know this much of it, that it is 
such a number as multiplied by itself will produce 12. 

It is easy to approximate to the square root of a number by 
taking a trial number and squaring it, when it will be at once 
seen whether such supposititious number is too great or too smalL 
It is also easy to find the square root by means of logarithms. 
But the ordinary arithmetical process for finding the square root 
is not difficult, and will be readily understood by one or two ex 
amples, 



SQUARE ROOTS OF NUMBERS. 47 

Tlius, in extracting the square roots of 15,625 and 998,001, 
the mode of procedure is as follows : — 



15625(125 998001(999 

1 81 



22)56 189)1880 

44 1701 



245)1225 1989)17901 

1225 17901 



Here, in the first place, wo separate the numbers into groups 
of two figures each, beginning at the right, by making a short 
line over each pair of figures, or by pointing them off into groups 
by such point or mark as shall not be confounded with the deci- 
mal point. Yv r e then find the next lowest square of the first 
group, which we set under that group, and subtract as in long 
division, setting the quotient in the usual place according to the 
mode of procedure in that process. TVe next double the quotient 
for the next trial divisor, and the quotient which we think we 
shall obtain we also place in the divisor, of which it forms a con- 
stituent part; and dividing by the divisor thus increased, we 
perform the division, setting the quotient in the usual place as in 
long division. "We then subtract, and for the next trial divisor 
we use the first term of the last divisor, and double the last term 
of the quotient. In the first example, consisting of five figures, 
we have only one figure in the first group, and that figure is 1. 
Now the square root of 1 is 1, which number w r e set in the quo- 
tient, and double it for the next trial divisor, which therefore 
becomes 2 ; and as 2 will go twice in 5, we set 2 in the quotient, 
and also, add it to the trial divisor to make the true divisor ; 
and so on. In the second example, the first group consists of 
the figures 99, the nearest square to which is 81, and we there- 
fore set 9 in the quotient, and put twice 9, or 18, for the next 
trial divisor, and we see that the number to be added thereto to 
exhaust the dividend must be large, as 18 is contained 10 times 
in 188. The number to be added to the trial divisor we find to 



48 ARITHMETIC OF THE STE AM-EXGESE . 

be 9, and we set it in the quotient^ and double it to add to the 
first trial divisor to form the second trial divisor; and so on 
through all the terms, bringing down at each stage a group of 
two figures, instead of a single figure, as in long division. "When 
there is a remainder after all the figures have been brought down, 
the number is not a complete square, and its exact root cannot 
be found, but it may be approximated to by using decimals to 
carry on the division with sufficient nearness for all useful pur 
poses. 

OX THE CUBES AXD CUBE BOOTS OF XUiTBEES. 

When any number is multiplied twice by itself, or, what is 
the same thing, when the square of a number is multiplied by 
the number, the product is the cube of the number. Thus 
2 x 2 x 2=8, and 8 therefore is the cube of 2. Also 4 is the square 
of 2, and 4x2=8. In like manner, 3x3x3=27, and 27 is the 
cube of 3 ; 4 x 4 x 4=64, and 04 is the cube of 4 ; ax ax a=a r> , 
and « 3 is the cube of «; or a-xa=aP. The cubes of the first 
nine numerals are 1, 8, 27, 64, 125, 216, 343, 512, anc 1 T29, and 
the respective differences of these numbers are 7, 19, 37, 61, 
127, 169, 217, 271, where we do not discern any law of increase. 
But if we take the respective differences of these last numbers, 
we obtain the numbers 12, IS, 24, 30, 36, 42, 48, 54, 60, where 
it is evident that the addition of the number 6 to each successive 
term produces the next one. 

In the cubes of fractions the same law holds as in the case 
a£ the squares of fractions. Thus as the square of ■§- is J, so the 
cube of % is -§•. So also -^ is the cube of ■&; -^t ^ s t ^ Le cube off, 
and f } is the cube of f . 

In the case of the cubes of mixed numbers, we first reduce 
those mixed numbers to an improper fraction, and then cube 
them as above. Thus the cube of 1^- is the same as the cube of 
f, which is -Y L or 3f, and the cube of 3£ or -^ is -g-f- 7 * or 34|4- 

The ctrbe of a d is a ?> J 3 , whence we see that if a number has 
factors, we may find its cube by multiplying together the cubes 
of the factors. Thus the cube of 12 is 1728. But 12 is com- 
posed of the factors 3 and 4; and the cube of 3 is 27, and the 



CUBES AND CUBE ROOTS OF NUMBERS. 49 

cube of 4 is G4. Hence 27 x 64=1728 will be the cube of 12, as 
by multiplying 12 by itself twice it is found to be. The cube of a 
positive number will always be positive, and of a negative num- 
ber, negative. This is obvious on considering that +ax +ax 
-\-a—-t-ct?, and that -«x-«= + a ! , and this multiplied again 
by — >« produces — « 3 . So the cube of — 1 is — 1, the cube of —2 
is —8, the cube of —3 is —27, and so of all negative numbers. 

The cube root of a number is expressed by the sign %/, and it 
is easy to determine the cube root of a number when the num- 
ber is really a cube. Thus we see at once that the cube root of 
1 is 1, that the cube root of 8 is 2, that the cube root of 27 is 3, 
that the cube root of 64 is 4, and that the cube root of 125 is 5. 
"We further see that the cube root of ^ 8 T will be f, of ff will be 
f , and of 2^, or |-f- , is f- or 1|-. But if the proposed number be 
not a cube, it cannot any more than in the case of the square 
root be expressed accurately, either by whole or fractional num- 
bers, though an approximate expression may be obtained that 
will be sufficiently near the truth for all useful purposes. For 
instance, 43 is not a perfect cube, and it is impossible to specify 
any number, whether whole or fractional, which, multiplied' by 
itself twice, will produce 43. If we take a number as nearly as 
we can to that which we suppose the cube root should be, and 
multiply it twice by itself, we shall at once see whether such 
trial number is too great or too small. Thus if we fix upon 3-| 
or f as the trial number, then we find that the cube of f- being 
2-f- 3 -, or 42|-, the number will err in defect, 42f- being -| less than 
43. By taking other numbers, we may approximate still more 
nearly to the true root, but we shall never be able to express it 
in figures precisely, and — as in the similar case in the doctrine 
of square roots — such quantities are termed irrational quantities. 

OX POWEES AXD BOOTS IX GENEEAL. 

The product arising from multiplying a number once or many 
times by itself is termed a power. The square of a number is 
sometimes called its second power ; the cube is sometimes called 
its third power, and we may have its fourth power, its fifth 
power, or any power depending on the number of the multipli- 
3 



50 



AEITHIIETIC OF THE STE AM-EXGIXE . 



cations, or we may say that the number has been raised to the 
second, third, fourth, or fifth degree. The fourth power of a 
number is sometimes called its diquadrate, but after this degree 
powers cease to have any other than numerical appellations. 

It is difficult to make the reason or processes of the ordinary 
arithmetical rule for the extraction of the cube root very intelli- 
gible without the aid of Algebra, of the processes of which the 
rile is only a translation. But an example will show the mode 
of procedure. 

Let us suppose that we had to extract the cube root of the 
number 80,677,568,161. 





4800 
369 


80677568161 
64 


123 


16677 




5169 


15507 


1292 


554700 

2584 


1170568 




557284 


1114568 


12961 


55987200 
12961 


56000161 




56000161 


56000161 



Here we first divide the number, beginning at the right hand, 
into groups of three figures in each — -just as in extracting the 
square root we divide the number into groups of two figures in 
each. In the last of the groups we thus form there happens, in 
this example, to be only two figures, and sometimes there will 
be only one. 

TTe now consider what is the next lower cube to the number 
80, and we find that it is 64, which is the cube of 4. "We set 
the figure 4 in the quotient, and subtract its cube 64 from 80, 
which leaves a remainder of 16. "We next bring down the fol- 
lowing period 677. 



MODE OF FINDING THE CUBE ROOT. 51 

The next step is to set the triple of the first figure of the root 
(12) at some distance to the left of the remainder. (There is 123 
in the sum, but the 3 will he accounted for presently.) We then 
multiply this triple by the first figure of the root, and place the 
product 48 between the 12 and the remainder, annexing two 
ciphers to it. 

We now divide the remainder by this 4800, as a trial divisor, 
and set the quotient 3 as the second figure in the root, and also 
after the 12, making 123. We next multiply this 123 by 3, the 
second figure of the root, set the product 369 under the 4800, 
and add them together. The resulting sum, 5169, is the first 
real divisor. We next multiply the divisor by the second figure 
of the root, and subtract the product 155 07, as in long division, 
bringing down the next period 568. 

To obtain the next real divisor we proceed as follows :— We 
first triple the last figure 3, of 123, which gives 129. (There is 
1292 put down, but the last figure, 2, wiU be accounted for pres- 
ently.) The other quantity, 5547, is found by adding 9, the 
square of the second figure of the root, to the two preceding 
middle lines, 369 and 5169. We now add two ciphers and re- 
peat the whole process, and we find the next figure of the root 
to be 2, which is the 2 added to the 129. 

In the case of decimals occurring in any number of which we 
have to extract the cube root, the distribution of the figures into 
groups of three each will begin at the decimal point, and will 
proceed to the left for integers, and to the right for fractions- 
adding ciphers where necessary to make up the required number 
• of figures. Thus if we had to extract the cube root of -01, we 
might write the number -010, and in like manner 24-1 might be 
written 24*100 

It will now be shown that to add the exponents of numbers 
■s equivalent to multiplying the numbers. 

ON ECOTS AS EEPKESENTED BY FEACTIONAL EXPONENTS. 

The multiplication or division of numbers is indicated by 
adding or subtracting their exponents, and as 2 may be written 



52 AEITH1IETIC OF TFE STEA3I-EXGLSE. 

1 1 

as 2 1 , then 2- x 2*=2 1 , since i+|- = l. As, soo, the third, 
fourth, fifth, &c, powers of a numher are represented by the 
expressions 2 3 , 2 4 , 2' 1 , &c, so the third, fourth, fifth, &c, roots 
are represented by the expressions ^/2, ^/2, ^/2, &c. The square 
root may he Trritten ^/, or more simply y/. Now as we have 

seen jhat 2 2 x2 3 = 2, and as ^/2 x ^/2 also ==2, it follows that 

i ii 

2* is another form of expression for v ; 2. So also 2 :i = ^'2, 2 + = 

i 
^/2, 2 5 = ^/2; and so of all other roots whatever. Since also 

2 x 2 7 =2 , =2 r , it follows that 2 ? is the same as V- : . In like 
manner, 2 r? = ^2 a and 2 T = ^'2 3 . 

"When the fraction which represents the exponent exceeds 
unity, it may either he expressed in the form of an improper 
fraction, or in that of a mixed number. For example, the frac- 

5. ol oL 

tion 2 2 may be expressed in the form 2*"-'. But 2"- is the prod- 

i 
uct of 2- by 2-, and it may be written in the forra 2 ^2\ 

OX THE CLASS OF FEACTIOXAL EXPOXEXT3 TEEMED LOGARITHMS. 

Since the square root of a given number is a number whose 
square is equal to that given number, and since the cube root of 
a given number is a number whose cube is equal to that given 
number, and so of all roots whatever, it follows that any number 
whatever being given, we may always suppose such roots of it 
that, raised to their respective powers, they shall always be equal 
to the given number. Since, also, powers with negative expo- 
nents are fractions, and powers with positive exponents are 
whole numbers, and as all numbers whatever may be expressed 
by whole numbers and fractions, it is clear that if we take any 
given number, such as 10, we may raise it to such a power either 
positive or negative as will make it equal to any number what- 
ever that we may think proper to assign. Thus if we fix upon 
the number 4, it is certain that there is a certain power of the 
number 10, which is equal to 4. Or if we fix upon the number 
40, or 47, or 57, or 881, or any other number whatever, then 



NATURE OF LOGARITHMS. 53 

there will be some power or other of 10 that will be equal to 
those several numbers. Putting b for this unknown exponent, 
then 10 =381, or any other number depending on the value of 
b. If instead of 10 we write the letter «, and instead of 381, or 
a raised to the power Z>, Ave write the letter c, then we obtain 
the expression a = c. Here c is the given number, a is the root 
or radix, and b is the exponent or logarithm of the number e 
with the radix a. The radix of the common system of logarithms 
is the number 10, and the logarithm of a given number is the 
power to which 10 must be raised to be equal to that given num- 
ber. Every number whatever has its corresponding logarithm ; 
and when we know its logarithm, we may, instead of the num- 
ber, use the logarithm, with this conspicuous advantage, that 
when we have to multiply two numbers together we shall ac- 
complish that end by adding their logarithms to obtain a new 
logarithm, the number corresponding to which will be the cor- 
rect product of the two numbers ; or if we have to divide one 
number by another, we shall accomplish the object by subtract- 
ing the logarithm of the one from that of the other — the differ- 
ence constituting a neAV logarithm, which will be the logarithm 
of the quotient. This quality of logarithms is apparent when we 
recollect that they are all exponents of a given number «, and 
that a- x a 3 = a 5 , or that a 5 x a 8 =a lz , where the multiplication is 
signified by adding the exponents. So also as a- x ' 3 = a% a 3X3 = 
a 9 , « 3X4 =a 12 , « 4X5 =a-°, it follows that to multiply a logarithm 
by 3, 4, 5, or any other number, is equivalent to raising the 
number to the third, fourth, fifth, or other corresponding power; 
and contrariwise, to divide the logarithm by 3, 4, 5, or any other 
number, is equivalent to the extraction of the third, fourth, fifth, 
or any other root of the number. From these considerations it 
will be at once apparent that by the use of logarithms an enor- 
mous amount of labour may be saved in performing arithmetical 
computations, and to facilitate such computations the logarithms 
of all the numbers usually occurring in calculations have been 
ascertained and arranged in tables, so as to facilitate then em- 
ployment. All positive numbers, such as 1, 2, 3, 4, 5, &c, are 
logarithms of the root or radix «, and of its positive powers, and 



5i ARITHMETIC OF THE STEAM-ENGINE. 

are consequently logarithms of numbers greater than unity. On 
the contrary, the negative numbers — 1, — 2, — 3, — 4, — 5, &c, 

are the logarithms of the fractions . — , — , — , . — , &c, which are 

a a 2 a 1 a 1 

less than unity and greater than nothing. Now as every signifi- 
cant number can omy be positive or negative, and as the loga- 
rithms of numbers greater than unity are positive, and the 
logarithms of numbers less than unity but greater than nothing 
are negative, there is no sign left to express numbers less than 
nothing, or negative numbers, and we must therefore conclude 
that the logarithms of negative numbers are impossible. 

It has already been stated that in the logarithmic tables at 
present in common use, the radix, of which the logarithmic num- 
ber is the exponent, is 10. If we denote this radix by «, then 
the logarithm of any number c is the exponent to which Ave 
must raise the radix a or 10, in order that the power result- 
ing from it may be equal to the number c. If we denote the 
logarithm of c by log. c, then 10 lo s- c =c. ISTow as a°=l and 
a l =a, so 10°=1 and 10'=10. But ?« the exponents are the 
logarithms of the numbers, it follows that the logarithm of 1 is 
0, and the logarithm of 10 is 1. So also log. 100 or 10-=2 ; log. 
1000 or 10 3 =3; log. 10000 or 10 4 =4; log. 100000 or 10'=5, and 
log. 1000000 or 10 6 =6. In like manner log. T V = —1; log. 

_1_ — 9 • 1nn» ? — 3 • lnc 1 — ^A- • Incr * ^ • 

ioo — - 1 ? lw b' iooo — u i ^g- i o o o o — ^ i iV & - iooooo — " i 

log. tottIoot = — 6 ; and so on indefinitely. 

Since log. 1=0 and log. 10=1, it is plain that the logarithms 
of all numbers between 1 and 10 mast be less than unity and 
greater than nothing. Let us suppose that it was required to 
determine the logarithm of the number 2. If we represent this 
logarithm by the letter x, then we shall have this expression 
10*=2. In order to determine the value of a?, we may make a 
few approximate suppositions. If we suppose x to be -J, we shall 

have 10 ¥ =2, which is manifestly too great, since 9 3 =3 and 10* 
must therefore be more than 3. If we suppose x to be |-, the 
quantity will still be too great. For if 10 ff =2, then 10 3 =2 3 , or 
10 1 or 10=8, which shows that -£ is too much. If we take \ as 



MODE OP COMPUTING LOGARITHMS. 55 

the exponent, then we have 10^=2, or 10 4 =2 4 , or 10=16, winch 
shows that £ is too small, while £ is too great. 

By pursuing the investigation in this manner, we should find 
with any required degree of accuracy what the exponent would 
be that, if 10 were raised to that power, would he equal to 2. 
This exponent or logarithm, as it is termed, would in point of 
fact he 0-3010300, or a little less than £, and in the logarithmic 
tables in common use the logarithms are always expressed in 
decimal fractions, as being the most convenient form for pur- 
poses of computation. The value of this decimal expressed in 

VtUgar iractions IS f o+TW + ToFo +TooF7 + T7rooW + TWo~froo + 

Too/flooiT' Logarithmic tables are commonly computed to seven 
places of decimals, as decimals carried to 1 places, though not 
expressing the result with absolute exactness, will, it is con- 
sidered, give results that are sufficiently accurate for all ordinary 
purposes. According to this method of expressing logarithms, 
the logarithm of 1 will be 0*0000000, since it is really =0. The 
logarithm of 10 will be 1-0000000, since it is=l. The logarithm 
of 100 will be written 2-0000000, =2, and so on. The logarithms 
of all numbers intervening between 10 and 100, and conse- 
quently composed of 2 figures, will be greater than 1 and less 
than 2, and are expressed by 1 + a decimal fraction. Thus log. 
50=1-6989700. The logarithms of numbers between 100 and 
1000 are expressed by 2 + a decimal fraction ; the logarithms 
of numbers between 1000 and 10,000 are expressed by 3+ a 
decimal fraction. The logarithms of numbers between 10,000 
and 100,000 are expressed by 4 and a decimal fraction, and the 
number prefixed to the decimal will always be 1 less than the 
number of figures in the given number. Thus the logarithm of 
2290 is 3-3598355, for as there are four figures in 2290, the num- 
oer prefixed to the decimal will be 3. The number prefixed to 
the decimal, or the integral part of the logarithm, is termed the 
cliaracteristic ; and when a number consists of four figures, such 
as the number 2290, its characteristic is invariably 3. If the 
number be reduced to 229, its characteristic will be 2 ; if reduced 
to 22 its characteristic will be 1, and if reduced to 2 its charac- 
teristic will be 0. There are therefore two parts to be con- 



5b ARITHMETIC OF THE STEAM-ENGINE. 

sidered in a logarithm: first the characteristic, which we can at 
once determine when we know the number of figures of which 
the given number consists ; and second the decimal fraction, 
which is determined by the nature of those figures. So also we 
know, at the first sight of the characteristic of a logarithm, what 
is the number of figures composing the number of which it is 
the logarithm. If for example the logarithm 6'4771213 be pre- 
sented, we know at once that the number of which it is the 
logarithm must consist of 7 figures, and must be over 1,000,000. 
The integral part of a logarithm therefore being so easily found, 
the main part requiring consideration is the decimal part, and it 
is that part alone which is given in the logarithmic tables in 
common use. To show the manner of using these tables, we 
may multiply together the numbers 343 and 2401 by the aid of 
logarithms. Here — 

Log. 343=2-5352941 ) , , , 
Log. 2401 = 3-3803922 j" ae * 

5-9156863 their sum. 
Log. 823540 = 5-9156847 nearest tabular log. 



16 difference. 

T7e look in the table of logarithms opposite the figures 343, 
and we find the number 5352941, which we know constitutes 
the fractional part of the logarithm, while the integral part will 
be 1 less than the number of figures in 343, or in other words 
the integral part will be 2. In like manner we find the logarithm 
of 2401, and adding these logarithms together, we find their 
sum to be 5-9156863. "We then look in the table to find the next 
less logarithm to this, which we find to be 5*9156847. We see 
at once by the magnitude of the characteristic that the number 
of which this is the logarithm must consist of six figures, and 
we find the number answering to this logarithm to be 823540. 
The difference between the logarithm formed by the addition of 
the two original logarithms and its next lower tabular logarithm 
js 16, and in the tables there is a column of differences intended 
to fix the numerical value of such differences, and which in this 



COMPUTATION OF COMPOUND QUANTITIES. 57 

case would amount to the number 3. "With this correction the 
product of 343 and 2401 will become 823543. 

It is in the extraction of roots, however, that logarithms be- 
come of the most eminent service. If, for instance, we had to 
extract the square root of 10, we should only have to divide 
the logarithm of 10 which is 1*0000000 by 2, which gives 
0*5000000 as the logarithm of the root required ; and by refer- 
ring to the table of logarithms, we should find that the number 
answering to this logarithm was 3*16228, which consequently is 
the square root of 10. So also if we had to extract the fifth 
root of 2 Ave should divide the logarithm of 2, which is 0*3010300, 
by 5, which gives a quotient of 0*0602060, the number answer- 
ing to which in the tables is 1*149 V, which consequently is the 
fifth root of 2. 

ON THE COMPUTATION OF COMPOUND QUANTITIES. 

Hitherto our investigations have been restricted to the modes 
of calculation suited to the measurement of simple quantities; 
but many of the quantities with which we have to deal in engi- 
neering practice are compound quantities made up of simple 
quantities in different forms of combination, and it is now neces- 
sary to consider the mode of computing the values of these 
compound quantities. One of the most familiar forms of a 
compound quantity is a sum of money expressed in pounds, 
shillings, and pence, or in other coins of different values. An- 
other variety is a given weight expressed in tons, hundred- 
weights, quarters, and pounds, or in other different hinds of 
weights. If we wish to know what number of pence there is in 
a sum of money, or what number of pounds or ounces there is in 
a given weight, the operation is termed reduction, and is per- 
formed by multiplying the given quantity by the number which 
shows how many of the next lower denomination makes one of 
the higher. Thus if we wish to know how many pence there 
are in 37?., we first multiply the 37?. by 20, which will show the 
number of shillings there are in 37?., for as there are 20 shillings 
in ]?., there will be 20 times'. 37 in 37?. "Now 37x20=740 
3* 



58 ARITHMETIC OF THE STEAM-ENGINE. 

shillings, and as there are 12 pence in 1 shilling, there will bo 
12 times 740=8880 pence in 37Z. If the sum were 37Z. 16s. 8d. 
in which we wished to find the number of pence, it is clear that 
the number of pence in 16s. 8d. must be added to the number 
already found. Now as there are 12 pence in 1 shilling, 12 
times 16=192, the number of pence in 16 shillings, to which, if 
we add the 8 pence remaining, we shall have 200 pence to add 
to the 8880, or in other words we shall have 9080 pence as the 
answer. So if we wish to ascertain the number of pounds 
weight in 3 tons, we have first to ascertain by a reference to a 
table of weights how many pounds there are in the ton, and 
which we shall find to be 2240. This number multiplied by 3 
will obviously be the number of pounds weight contained in 3 
tons. But if the weight which we were required to find the 
number of pounds in were 3 tons 7cwt. 2 quarters and 8 pounds, 
we should first have to multiply the 3 tons by 20 to reduce them 
to cwts., and as there are 20 cwt. in the ton, 3 tons would he 
60 cwt. But besides these we have 7 cwt. more, so that we 
have in all 67 cwt. Now as there are 4 quarters in the cwt., 
there will in 67 cwt. he 4 times 67=268 quarters, to which we 
have to add the two quarters of the original sum, making in all 
270 quarters in the weight. But as there are 28 lbs. in 1 quarter, 
there will he 28 times 270=7560 lbs. ; and as there are 8 pounds 
besides to be added, the sum total of the weight will be 7568 lbs. 
So if we wished to know how many square inches there were in 
2J square feet, it is plain that as there are 144 square inches in the 
square foot, there will he 288 square inches in 2 square feet, and 
36 square inches in i of a square foot, and 288 + 36=324 square 
inches. In performing these and similar operations it is of course 
necessary to have access to proper tables of weights and measures, 
or, in other words, to certain standard magnitudes, as it is impossi- 
ble to form an idea of any magnitude except by comparing it 
with some other magnitude, such as a pound, a foot, or a gallon, 
of which we have a definite conception. 

On tlie addition of compound quantities. — The first step in 
performing this addition is to set the quantities to be added un- 
der one another, so that terms of the same kind may he in the 



£ 


s. 


& 


13 





8 


2 


5 


6 


23 


4 


1 


3 < 


8 


10 


12 


9 


7 





13 


4 


89 


2 


6 



COMPUTATION OF COMPOUND QUANTITIES. 59 

same column. When the relation between the different quanti- 
ties is known — as it is in all cases of arithmetical addition — we 
add up the numbers in the right-hand column, and divide by the 
number in this column which makes 1 in the next column. "We 
then set the remainder, if any, under the first column, and carry 
the quotient to be added to the next, and so on through all tho 
columns. Thus in adding up the pounds, shillings, and pence 
here set down we proceed as follows : 

We first arrange the pounds, shillings, and pence in three 
columns, with the units under the units, the tens un- 
der the tens, and so on, as in simple addition. We 
then add up the column of pence, and find how many 
pence it contains. But as every group of 12 pence 
makes 1 shilling, we divide the total number of pence 
by 12 to find how many of such groups there are, or, 
in other words, how many shillings there are in the 
total number of pence. These shillings we transfer 
to the shillings column, and as after we have done this there are 
6 pence left, we write the 6 beneath the pence column, and 
then proceed to add up the shillings, beginning with the number 
of shillings we have brought from the pence column. Having 
thus ascertained the total number of shillings, we find how 
many pounds there are in that number of shillings by dividing 
by 20, there being 20 shillings in the pound sterling ; and after 
having found this number of pounds, we carry it to the pounds 
column, and the 2 shillings which we find remaining we write 
under the shillings column. We then proceed to add the pounds 
column, beginning with the number of pounds in shillings which 
we have carried from the shillings column. 

In adding up cwts., quarters, and pounds, the mode of pro- 
cedure is precisely the same, only as there are 28 lbs. in 1 quar- 
ter, 4 quarters in 1 cwt., and 20 cwt. in 1 ton, the divisors we 
use at each step must vary correspondingly. This will be plain 
from the following example : 

Here we find the sum to be 20 cwt. 3 qrs. and 17 lbs., or 1 
ton cwt. 3 qrs. and 17 lbs. ; for, after adding the first column, 
and dividing the sum by 28, we have 17 left, and after add- 



GO AEiimiETIC OF THE STEAM-ENGHSE. 

cwt er. lbs. i&g the second or quarters column with the 

3 3 12 addition of the number of quarters in lbs. that 

2 3 18 we nave carried over from the lbs. column, we 

6 2 19 divide the number so obtained by four to obtain 

2 the number of cwts. there are in all these quar- 

1 ton 3 17 * er5, ^~ e can T the cwt. so obtained to the 

===== cwts. column, and write beneath the quarters 

column the 3 quarters which we find are left. Proceeding in 

the same way with the cwts. column, we find its sum to be 20 

cwts. or 1 ton ; and the total quantity to be 1 ton cwt. 3 qrs. 17 

lbs., as stated above. 

Subtraction of compound, quantities. — "When we wish to 
subtract one compound quantity from another, we write the less 
under the greater, so that the terms of the same kind may be in 
the same column, as in the case of addition. "We then subtract 
the right-hand term of the lower line froni that of the npper, if 
possible. But if this cannot be done, we must transform a unit 
of the next higher term into its equivalent number of units of 
the first term, and then performing the subtraction, we write 
the difference under the first column, and we increase by 1 the 
next term to be subtracted to compensate for the unit previously 
borrowed. In algebra, the usual process of subtraction is to 
change the signs of the lower line, and then to proceed as in 
addition. 

If we had to take 277. 8*. 4%d. from 347. Vis. 9f/7., we should 

write down the greater sum first and the less under it, so that 

£34 17 9* pounds should fall under pounds, shillings under 

£lT 8 4i sMliings, and pence under pence. Taking \d. from 

,._ 7 ~ id. we have id. over, which we write down, and 

i. i 9 5^ 

= then taking 4c7. from Pc7. we have od. over, which 
we also "write below the column of pence. Next taking 8a. 
from 17s. we have 9s. left, and taking 77. from 1-47. we have 7 7 .. 
and carrying 1 to the 2 appearing in the next place we have 3 
from 3, which leaves nothing. The difference, therefore, be- 
tween 347. 17s. 9%d, and 27". 8*. 4£d. is 11. 9*. o±d. If we had to 
subtract 227. 18*. Il-fc7. from £37. 6. 0\d., we should proceed 
thus :— 



COMPUTATION OF COMrOUND QUANTITIES. 61 

£23 6 0+ Here taking %d. from \d. we have to borrow 

£22 IS llf Id. or 4 farthings from the next term, and we have 

"~7~ ~ 7! then 6 farthings to he suhtracted from, and \d. 

£0 7 0?- J* 

subtracted from £ d. leaves %d. In the next term 



we have llf7., which must be increased to Vld. on account of 
the penny before borrowed ; and as we have no pence to sub- 
tract from we must borrow Is. from the next term, and chango 
it into 12 pence, and 12 pence taken from 12 pence leaves noth- 
ing. In the next term of shillings we have 18, which must be 
increased to 19 in consequence of the previous borrowing of Is, 
to carry to the column of pence, and 19s. taken from 11. 6s. or 
26s. leaves 7s. In the next term the 2 has to be increased to 8 
to make up for the 11. imported into the column of shillings, 
and 23 taken from 23 leaves nothing. The difference between 
these two sums is consequently Is. OfcZ. 

If we have to take 5 tons 12 cwt. 3 qrs. 27J lbs. from 93 

tons 8 cwt. 1 qr. 6 lbs., we proceed as follows : 

tons cwt or. lbs. ^ ere i ^ D - taken from 1 lb. leaves \ lb., and 

93 8 ' 1 6' 28 lbs. taken from 1 qr. and 6 lbs. or 34 lbs., 

. 5 12 3 27J - leaves 6 lbs. Then 4 qrs. taken from 1 cwt. and 

87 15 1 61 1 qr. or 5 qrs. leaves 1 qr. ; and 13 cwt. taken 

from 1 ton and 8 cwt. or 28 cwt. leaves 15 cwt. 

Lastly, 6 tons taken from 93 tons leaves 87 tons. 

If we wish to subtract 6—2 + 4 from 9—3 + 2, we may either 
perform the subtraction by first adding the quantities together, 
and then subtracting the sum of the one from that of the other, 
or we may change the signs of the quantity to be subtracted, 
and then add all together, which will give the same result. 
Thus 6— 2 = 4, and 4+4= 8. So also 9—3 = 6, and 6 + 2 = 8, 
Subtracting now one sum from the other, we get 8—8 = 0. 
But if we change the signs of 6—2 + 4, and add it to 9—3 + 2, 
we have 9— 3 + 2— 6 + 2—4 = 0. 

Multiplication of compound quantities. — When we wish to 
perform the multiplication of any compound number, such as 
pounds, shillings, or pence, or hundredweights, quarters, and 
pounds, we set the multiplier under the right-hand term of the 
multiplicand, multiply that term by it, and find what number 



62 ARITHMETIC OF THE STEAM-ENGINE. 

of times one of the next higher term is contained in the prod 
net, "which number is tc be carried to the next term, "while the 
remainder, if any, is to be written nnder the right hand or low- 
est term. ~We> must then multiply the next term in like man- 
ner, and so until the whole have been multiplied. Thus if we 
had to multiply 23?. 13s. 5d. by 4, we should proceed as follows : 
Here we first multiply the pence, and 4 times 5 
°"~ 4 pence is 20 pence, which is Is. Sd. ; and so we put 

down 8 and carry 1. In the shillings term we say 



^ 94 13 8 4 times 3 are 12, and with the addition of the 1 
shilling brought over from the pence term, the 12 
becomes 13. Then 4 times 10 is 40 shillings, which make just 2 
pounds, so we carry the 2 pounds to the pounds place, leaving 
the 13 previously obtained in the shillings place. Proceeding to 
the pounds, we say 4 times 3 are 12 and 2 are 14, and 4 times 2 
are 8 and 1 are 9. Hence the product is 94?. 135. 8d., which 
sum would also be obtained by writing down 23?. 13s. 5d. four 
times under one another, and ascertaining their sum by addi- 
tion. 

"When the multiplier is large, but is composed of two or more 
factors, we may, instead of multiplying by the number, multiply 
successively by it3 factors. Thus if we have such a sum as 
£23 lis. 4§ (1. to multiply by 36, then as 36 is a number repre- 
sented by the factors 6 x 6, 4 x 9 or 3 x 12, we shall obtain 
the same result by multiplying by any set of these factors as by 
multiplying by the 36 direct. Thus — 



£23 11 


4J 
6 


£23 11 


4 


£23 11 4| 
3 


141 8 


4? 
6 


94 o 


9 


10 14 2£ 
12 


£848 10 


3 


£848 10 


o 


£848 10 3 



In like manner if we had to multiply the sum £17 os. 0' 2 d. 
by 140, then as 140 is made up of the factors 7 x 20, or 
4x5x7, we may multiply by these numbers instead of the 



£17 


o 
O 


o* 






4 


68 


12 


2 
5 


343 





10 






7 


£2401 


5 


10 



COMPUTATION OF COMPOUND QUANTITIES. G3 

140. In cases however in which the multiplier 
cannot be broken up into factors, we must mul- 
tiply each term by it consecutively. Thus if 
£23 11* 4f& be multiplied by 37, we have first 
3 farthings multiplied by 37, which gives 111 
farthings or 27 pence and 3 farthings. "Writing 
down the 3 farthings and carrying the 27 pence, 
we have 37 times 4 pence or 148 pence, and add- 
ing the 27 pence we have 175 pence, which as 
there are 12 pence in the shilling we divide by 12 and get 14 
shillings and 7 pence. We set down the 7 in the pence place 
and carry the 14 to the shillings place, and we thus proceed 
through all the terms nntil the multiplication is completed. The 
same mode of procedure is adopted if, instead of pounds, shil- 
lings, and pence, w T e have hundredweights, quarters, and pounds 
or any other quantities whatever. 

Division of compound quantities.— In the arithmetical divi- 
sion of compound quantities, we set the divisor in a loop to the 
left of the dividend and divide the left-hand term by it, setting 
the quotient under that term. If there is any remainder we re- 
duce it to the next lower denomination, adding to it that term, 
if any, of the dividend which is of this lower denomination. 
We then divide the result by the divisor and so on, until all the 
terms have been divided. Thus if we had to divide £38 6s. 8f <rZ. 
by 3, we should proceed as follows : — 

r> „ 7 Here we find that 3 is contained in 3 once, and in 

k, S. a. ' 

3)38 6 8\ 8, 2 times and 2 over. But 2 pounds are 40 shillings, 

: and 6 are 46 shillings, and 46 divided by 3 gives 15 ■ 

12 15 6 a 

4 and 1 over, which 1 shilling is equal to 12 pence, 

and adding to this the 8 pence in the dividend, we have 20 pence 
to be divided by 3. JSTow 20 divided by 3 gives 6 and 2 over, 
which 2 pence are 8 farthings, and adding thereto the 1 farthing 
in the dividend, we have 9 farthings to divide by 3, or 3 far- 
things. It is clear that £12 15s. 6%d. multiplied by 3 will again 
give the £38 6s. 8±d. of the dividend. 

If we have to divide a number by 10, we may accomplish the 
division by pointing off one figure as a decimal, if by 100 we point 



64 ARITHMETIC OE THE STEAM-ENGINE. 

off two figures, if by 1000 three figures, and so on. Thus if wa 
have to divide £2315 14s. 7d. by 100, we may proceed as follows : 
Here we point off two figures of the highest 
term as decimals, which leaves £23. We.next mul- 
tiply the residual decimal by 20 to reduce it to 
shillings, bringing down the 14 shillings in the 
dividend, and we obtain 3 shillings and "14 of a 
shilling, which fraction we multiply by 12 to bring 
it to pence, and we bring down thereto the 7 pence 
in the dividend. We obtain as a product 1'75 pence, 
and multiplying in like manner "74 by 4 to bring it to 
farthings, we obtain 3 farthings, making the total quotient £23 
3s. lp. This sum multiplied by 100 will make £2315 14s. 7d. 
"When the divisor is large but may be broken up into factors, we 
may divide separately by those factors. Thus if we wish to divide 
£3702 3s. Qd. by 24, then as 24= 4 x 6 or 3 x 8 or 2 x 12, we 
may divide the sum by any pair of factors instead of by the 24. 



£ 


s. d. 


23-15 


14 7 


20 




3-14 




12 




1-75 




4 





£ s. d. 
4)3762 3 6 


£ s. 
3)3762 3 


d, 
6 


£ s. 

2)3762 3 


d. 
6 


6)940 10 10£ 


8)1254 1 


2 


12)1881 1 


9 


£156 15 If 


£156 15 


H 


£156 15 


H 



425 

423 



When the number cannot be broken up into factors we must 

proceed by the method of long division. Thus if we had to 

divide £3715 18s. 9c?. by 47 we should proceed as follows : — 

£ s. d. Here we find first how often 47 will 

^lll 5 i8 9 ^ 9 l 3 g'O in 371, and we find it wiU be 7 times, 

when we write the 7 in the quotient and 

multiply the divisor by it, setting the 

product under the first three figures of 

2 the dividend. Subtracting now the 329 
20 
from the 371, we find that the remainder 

58(1 is 42, and we bring down the next figure 

[ of the dividend and find how often 47 is 

11 contained in 425. We find that it is 9 

times, which completes the division of 

141(3 the pounds. The 2 pounds remaining 

we next multiply by 20 to bring them to 



141 



NATURE OF AN INFINITE SERIES. G5 

shillings, adding the 18 shillings of the dividend, which together 
make 58 shillings, the 47th part of which is 1 shilling and jiths 
over. Multiplying this by 12 to bring it to pence, and dividing 
by 47, we get 3 pence, which completes the operation. 

In cases where we have to divide a compound quantity by 
another of the same kind, such as money by money or weights 
by weights, the requirement is equivalent to that of finding 
what number of times the one amount is comprehended, in the 
other. IV e cannot of course divide a quantity by another of a 
different kind, as money by weight, nor can we multiply money 
by money or weight by weight. If we are required to divide 
such a sum as £3 Is. 6d. by 16s. 10|-<:Z., we reduce both the num- 
bers to the lowest denomination appearing in either, which in 
this case is half pence, and we then divide the greater number 
by the less. !N"ow £3 7s. 6&==1620 half pence and 16s. 10f& 
= 405 half pence, and 1620 -f- 405 =4. So if we had to divide 
3 tons 2 cwt. 2 qrs. 21 ibs. by 2 qrs. 7 lbs., then as the first 
amount is equal to 6993 lbs. and the second to 63, the question 
becomes one of dividing 6993 by 63, which we find gives 111. 
It follows consequently that 2 qrs. 7 lbs. multiplied by 111 =3 
tons 2 cwt. 1 qr. 21 lbs. 

As a square foot contains 144 square inches, we must, in as- 
certaining the number of square feet in any given number of 
square inches, divide by the number 144, and as a cubic foot 
contains 1728 cubic inches, we must, in ascertaining what num- 
ber of cubic feet there are in any number of cubic inches, divide 
oy the number 1728. So also there are nine square feet in a 
square yard, and 27 cubic feet in a cubic yard. A cubic foot 
contains very nearly 2200 cylindric inches or solid cylinders 1 
inch in diameter and 1 inch high ; 3300 spherical inches or balls 
1 inch diameter ; and 6600 conical inches or cones 1 inch diam- 
eter and 1 inch high. 

OS TUE EESOLTJTIOX OF FUACTIOXS IXTO IXFIXITE SEEIES- 

"We have already explained that in decimal fractions the de- 
crease at every successive figure is ten times, just as in common 



60 AEITHMETIC OF THE 5TEAM-EXGINE. 

numbers the increase at every successive number is ten times. 
Thus the number 666 means 600 + 60 + 6, so that the first 
figure by virtue of its position alone is ten times greater than 
the second, and the second by virtue of its position alone is ten 
times greater than the third. Precisely the same law holds 
when we descend below unity, as we do in every case in which 
the decimal point is introduced, as the meaning of the decimal 
f.oint is, that all the numbers to the right of it are less than 
unity, and that they diminish ten times at each successive figure, 
just as ordinary numbers do. The expression 666*666 therefore 
means six hundred and sixty-six with the addition of 6 tenths, 
six hundredths, and six thousandths, or, what is the same thing, 
of QQQ thousandths. The expression might therefore be written 
666 + T \+ T fo+ T o 6 o Tr or 666 T 8 o 6 oV Every decimal fraction may 
consequently be considered as a vulgar fraction, with a denom- 
inator of 10 or 100 or 1000 understood, according to the position 
of the decimal. Thus '1 is equivalent to T V, '01 is equivalent 
to T ^o", and *001 is equivalent to T oVo- Now the fraction ^ is 1 
divided by 3, and if we perform the division we shall have 

3)1-00000 



•33333, &c, 

and so on to infinity. The vulgar fraction 4 is consequently 
equal to the infinite series *33333, &c, which, at each successive 
term to which it is carried, becomes more nearly equal to the 
fraction of £, but never becomes exactly equal thereto. Any 
vulgar fraction may be at once converted into its equivalent 
decimal by dividing the numerator by the denominator, adding 
as many ciphers to the numerator as may be necessary to enable 
the division to be carried on. But some of the divisions thus 
performed, it will be found, may be carried on for ever, and such 
a series of numbers is termed an infinite series. As a visible 
exemplification of the continual approach of two quantities to 
one another without ever becoming equal, we may take the fol- 
lowing example : 

Here we have a line a b which we may divide into any rum- 



ARITHMETICAL EXAMINES. 



67 



Ler of equal parts, and we draw the line a o at rig-lit angles 
with a b : at o we draw another short line ao parallel to a b, and 
we set off the distance ca equal to a1. If now we draw the 
diagonal line la we shall cut off the half of a c, or shall bisect 
it in the point x. and by drawing the lines 2a, 3a, 4a, 5a, &c, 




we cut off successive portions of xc, and therefore continually 
diminish it. But we never can cut it all off, however extended 
we may make the line a b, and however numerous the addi- 
tional portions cut off may be. The quantity xq becomes more 
and more nearly equal to xa, the greater the length of the line 
a b, and the more numerous the fractional quantities successively 
cut off. But no extension of the operation short of infinity 
could make the portions cut off from xo equal to xa. 



ARITHMETICAL EXAMPLES. 



Having now illustrated with adequate fulness of detail the 
elementary principles of engineering arithmetic, it is only neces- 
sary that we should add some examples of the method of per- 
forming such computations as are most likely to be required in 
practice. 

Reduction. — This is the name given to the process of con- 
verting a quantity expressed in one denomination into an equiv- 
alent quantity expressed in another denomination, such as tons 
expressed in ounces, or miles in yards. 

Example 1. — Reduce 151. Is. 0%d. to farthings. 



68 ARITHMETIC OF THE STEAM-ENGINE. 

15 7 Of Here we first multiply the pounds by 20, there 

being 20 shillings in the pound, and Ave bring 

307s. down the 7 shillings, making 307 shillings. "Wo 

12 then multiply the shillings by 12, there being 

~~~ , 12 pence in the shilling, and here we have nc 

4 ' pence to bring down. Finally, we multiply by 



4, there being 4 farthings in each penny, and 

14>739 / - Am - we bring down the 3 farthings, making 14,739 
farthings in all. 

Example 2. — Eeduce 23 tons to pounds avoirdupois. 

By a reference to a table of weights and measures, we find 
that there are 2,240 pounds in the ton ; 23 times 2,240, there 
fore, or 51,520 lbs., is the answer required. 

Example 3. — Eeduce 100 square yards to square inches. 
Here, as each square yard contains 9 square feet, and each square 
foot 144 square inches, there will be 9 times 144 or 1,296 square 
inches in each square yard, and 100 times this, or 129,600 square 
inches, in' 100 square yards. It may be well here to remark that 
100 square yards is a very different quantity from 100 yards 
square, which would, in fact, contain an area of 10,000 square 
yards. 

Example 4. — Eeduce 7 cubic yards, 20 cubic feet, to cubic- 
inches. As there are 27 cubic feet in a cubic yard, there will 
be 27 times 7, or 189 cubic feet in 7 cubic yards, to which add- 
ing 20, we have 209 cubic feet in all; and as there are 1,728 
cubic inches in a cubic foot, we have 1,728 times 209, or 361,152 
cubic inches as the answer required. 

Quantities are brought to a higher denomination by the re- 
verse of the process indicated above, that is, by dividing, instead 
of multiplying. Thus, by dividing by 4, 12, and 20, it will be 
found that 14,739 farthings are equal to 15?. 7s. 0f<2. ; by .divid- 
ing 51,520 lbs. by 2,240, that the quotient is equal to 23 tons ; 
and by dividing 129,600 square inches by 144, and then by 9, 
that the result is 100 square yards. So also by dividing by 
1,728, it will be found that 301,152 cubic inches are equal to 
209 cubic feet, and dividing again by 27, we find the answer tc 
be 7 cubic yards and 20 cubic feet. 



EXAMPLES OP PRACTICAL COMPUTATIONS. 69 

Mexsijratiox of Surfaces and Solids. — The area of a rec- 
tangular surface is obtained by multiplying the length by the 
breadth. The area of a circle in circular inches is obtained by 
multiplying the diameter by itself; and the area of a circle in 
square inches is obtained by multiplying the diameter by itself, 
and by the decimal '7854. The circumference of a circle is 
3*1410 times its diameter. The capacity of a rectangular solid 
is obtained by multiplying together its length, depth, and thick- 
ness ; and the capacity of a cylinder in cubic feet or inches is 
obtained by multiplying the area of its cross section or mouth, 
expressed in square feet or inches, by its depth in feet or inches. 

Example 1. — What is the quantity of felt required to cover 
the side of a marine boiler that is IT feet 8 inches long, and 3 
yards high ? 

Here we first reduce the measurements to inches, and as 17 
ft. 8 in. is equal to 212 inches, and as 3 yards or 9 feet is equal 
to 108 inches, we have an area represented by 212 multiplied 
by 108 inches, or 22,896 square inches. Now, as there are 144 
square inches in each square foot, we shall, by dividing 22,896 
by 144, find that the area is 159 square feet, and dividing this 
by 9 to bring the quantity into square yards, we find that the 
area is 17 square yards and 6 square feet over. 

Since the area is obtained by multiplying the length by the 
breadth, it will follow that if we divide the area by the length 
we shall get the breadth, and if we divide the area by the 
breadth we shall get the length. 

Example 2. — "What is the weight required to be placed on 
top of a safety-valve 4 inches diameter, to keep it down until 
the steam attains a pressure of 20 lbs. on each square inch ? 

Here 4x4=16 circular inches, and 16 x -7854= 12'56S 
square inches, which x 20 the pressure on each square inch = 
251-32 lbs. 

Example 2. — The engine of the steamer 'Arrogant' is a 
trunk engine, in which the piston rod is widened into a hollow 
trunk or pipe 24 inches diameter, which correspondingly re- 
duces the effective area of the piston. As the cylinder is 60 
inches diameter, reduced by a circle 24 inches diameter, what 



70 ARITHMETIC OF THE STEAM-ENGINE. 

will be the diameter of a common cylinder to have an equa, 
area ? 

Here 60 3 x '7854 =- 2827*44 square inches, and 24 3 x '7854 = 
452*39 square inches, and 2827*44 diminished by 452*39 =— 2375*05 
square inches, This is as nearly as possible the area of a cylin- 
der 55 inches in diameter, which is 2375*83 square inches. 

Example o. — The steamer 'Black Prince' has two direct- 
acting* trunk engines, with cylinders equal to 104J- inches diam- 
eter, and the length of the stroke is 4 feet. The engines make 
55 revolutions per minute. "What will be the number of cubic 
feet of steam required per hour to fill the cylinder ? 

Here the diameter being 104J inches, the area of each cylin- 
der will be *7854 times 104J squared, or it will be 8835*7 square 
inches, or 61*3 square feet. As the piston travels backwards 
and forwards at each revolution, it will pass through 8 feet dur- 
ing each revolution ; and the volume of steam required by each 
cylinder in each revolution will be 8 times 61*3, or 490*4 cubic 
feet. As there are two engines, the total volume of steam re- 
quired in each revolution will be twice 490*4, or it will be 980'8 
cubic feet ; and as there are 55 strokes in each minute, the 
expenditure per minute will be 55 times 980*8, or 53,944 cubic 
feet. The expenditure per hour will, of course, be 60 times 
this, or 3,236,640 cubic feet. In all modern engines the steam 
is not allowed to enter the cylinder from the boiler during the 
whole stroke; and the expenditure of steam will be less the 
sooner it is cut off or prevented from entering the cylinder. 
But the cylinder, nevertheless, will still be filled with steam, 
though of a less tension, than if the supply from the boiler had 
not been interrupted ; and the space traversed by the piston will 
always be a correct measure of the steam consumed, taking that 
steam at the pressure it has at the end of the stroke. 

Example 4. — The ' Black Prince ' has an area of immersed 
midship section of 1,270 square feet ; or, in other words, if the 
vessel were cut across in the middle, the area of that part below 
the water would be 1,270 square feet. The diameter of the 
screw is 24 feet 6 inches, the nominal power is 1,250, and the 
indicated power 5,772 horses. What is the ratio, or proportion, 



EXAMPLES OP PRACTICAL COMPUTATIONS. 71 

of the area of midship section to the area of the circle in which 
the screw revolves ? and what is the ratio of the immersed mid 
ship section to the indicated power? 

Here the diameter of the screw being 24£ feet, the area of 
the circle in which it revolves will he 471*436 square feet, and 
1,2 TO divided by 471 '436 being 2-C9, it follows that the ratio of 
immersed midship section to screw's disc is 2*69 to 1. So, in 
like manner, the indicated power 5,772, divided by 1,250, gives 
a ratio of indicated power to immersed midship section of 4*54 
to 1. "With these proportions the speed was at the rate of nearly 
15 knots per hour, so that to ensure such a speed in a vessel 
like the ' Black Prince,' it is necessary that there should be 4^ 
or 5 indicated horse-power for each square foot of immersed 
midship section of the hull. 

Example 5. — If it were desired to encircle the screw of the 
' Black Prince ' with a sheet-iron hoop, what length of hoop 
would be required for the purpose ? 

The diameter of the screw being 24| feet, the circumference 
of the circle in which it revolves will be 3 "141 6 times 24|, or it 
will be 76-969 feet. 

Example 6. — A single acting feed pump has a ram of %\ inches 
diameter and 18 inches stroke, and makes 50 strokes per minute. 
How much water ought it to send into the boiler every hour ? 

Here the area of the ram will be 4*9 square inches, and the 
stroke being 18 inches, 18 times 4*9 or 88*2 cubic inches will be 
expelled at every stroke, supposing that there is no loss by leak- 
age or otherwise. As there are 50 strokes made in the minute, 
the discharge per minute will be 50 times 88*2, or 4,410 cubic 
inches; and there will be 60 times this, or 264,600 cubic inches 
discharged in the hour. As there are 1,728 cubic inches in the 
cubic foot, we get the hourly discharge in cubic feet by dividing 
264,600 by 1,728, and we shall find the discharge to be 153-125 
cubic feet. A cubic inch of water will make about a cubic foot 
of steam, of the same pressure as the atmosphere. 

Example 7. — A cubic foot of water weighs 1,000 ounces. 
What will be the weight of water in a vessel which is fi-Vssd to 
the brim, and which measures a yard each way ? 



i'l ARITHMETIC OF THE STEAM-ENGINE. 

As there are 27 cubic feet in a cubic yard, tbe weight re- 
quired will be 27,000 ounces, which, divided by 16, the number 
of ounces in a pound, gives 1,687 lbs. and 8 oz., and dividing 
again by 112, the number of lbs. in each cwt., we get 15 cwt. 
7 lbs. 8 oz. 

Example 8. — Two steamers being started together on a race, 
'X was found that the faster went 5 feet ahead of the other in 
each 55 yards : how much will she have gained in half a mile? 

As a mile is 1,760 yards, half a mile is 880 yards, and there 
are 16 times 55 yards, therefore, in half a mile. As in each 55 
yards 5 feet are gained, there will be 16 times 5 feet, or 80 feet 
gained in the half mile, or 26 yards 2 feet. 

Example 9. — The 'Warrior,' a steamer of 6,039 tons burden, 
and 1,250 nominal horse-power, attained a speed on trial of 
14*356 knots per hour, the engines exerting an actual power of 
5,469 horses. The screw was 24| feet diameter, and 30 feet 
pitch, or, in other words, the twist of the blades was such that it 
would advance 30 feet at each revolution, if the advance were 
made without any resistance. The engines made 54*25 revolu- 
tions per minute, and if the screw advanced 30 feet in each revo- 
lution, it would advance 1627*5 per minute, or 16*061 knots per 
hour. In reality, however, the screw only advanced through 
the same distance as the ship, namely, 14*356 knots per hour. 
The actual advance, therefore, was less than the theoretical ad- 
vance by 1*705 knots per hour, which difference is called the 
slip of the screw; for 1*705 added to 14*356 makes 16*061, 
which would be the speed of the vessel at this speed of the screw 
if there was no slip. 

Example 10. — What is the diameter of a piston of which the 
area is 2827*44 square inches ? 

Here 2827'44, divided by *7854,= 3600, the square root of 
which is 60. This is the diameter required. 

Example 11. — A cubical vessel of water weighs 5 tons, ex- 
cluding the weight of the vessel. What is the* length of the 
side? 

As there are 1,000 ounces in a cubic foot of water, we know 
that there will be the same number of cubic feet in the vessel as 



EXAMINES OF PRACTICAL COMPUTATIONS. 



73 



the number of times 1,000 ounces is contained in 5 tons. Now 
as there are 2,240 lbs. in the ton, there will be 5 times this, or 
11,200 lbs. in 5 tons, or 179,200 ounces. Dividing this bj 1,000, 
Ave have 179*2 cubic feet as the content of the vessel. 

To find the length of the side we must extract the cube root 

of 179-2. "We soon see that 



156 



7500 
936 

8436 



1683 



940800 
5049 

945849 



179-2(5-63 
125 



54-200 



50-616 



3584000 

_2837547 
~ 746453 



the root must lie between 5 
and 6, for the cube of 5 is 
125, and the cube of 6 is 216. 
Taking 5 as the next lowest 
root, we set this number as 
the first figure of the quotient 
and subtract its cube as in 
long division, bringing down 
three more figures at each 
stage, and here two of these must be ciphers. 

We now triple the root 5, and set down the 15 to the left, 
and we multiply this triple number by the first figure of the 
root making 75, which number we set down between the 15 
and the remainder, adding two ciphers to it, which make it 
7500. We now consider how often the trial divisor 7500 will 
go into the remainder 54200, after making some allowance for 
additions to the divisor, and we find it will be 6 times. We 
place the 6 as the second figure of the root, and we also place it 
after the 15. We multiply the 156 by the 6, and place the prod- 
uct under the 7500. The resulting number, 8436, is the first 
true divisor. 

We now bring down the next period of three figures, and as 
there are no figures remaining to be brought down, we introduce 
three ciphers. We triple the last figure of 156, which gives 168, 
and we add the square of 6, which is 36, to the sum of the two 
last lines, 936 and 8436, making in all 9408, to which we add 
two ciphers, making 940800, and we then see how often this 
sum is contained in 3584000. We find that it will be 3 times, 
and we set down the 3 as the next figure of the root, and also, 
after the 1G8, making 1683, and we add three times this to the 
940,800, making 945849, which is the second real divisor. We 
4 



74 ARITHMETIC OF THE STEAM-ENGINE. 

hot?" multiply this divisor by the last figure of the quotient, and 
subtract the product as in long division, leaving as a remainder 
746453, to which, if we wished to carry the answer to another 
place of decimals, we should annex three ciphers, and proceed 
as before. For all ordinary purposes, however, an extraction to 
the second place of decimals is sufficient, and if we cube 5 - 63, we 
sli all find the resulting number to be 178*453547, being a little 
le3s than 179-2. 

Example 12. — The density or specific gravity of mercury ia 
13*59 times greater than that of water, and the specific gravity 
of water is 773*29 times greater than that of air of the usual at- 
mospheric pressure. What will be the height of a column of 
water that will balance the usual barometric pressure of 30 
inches of mercury, and what also will be the height of a column 
of air of uniform density that will be required to balance that 
pressure ? 

Here the mercury being 13*59 times more dense than the 
water, or in other words, the water being 13*59 times more light 
than the mercury, it will be necessary that the height of the 
column of water should be 13*59 times greater than that of the 
column of mercury, in order to balance the pressure. If, there- 
fore, the column of mercury be 30 inches high, the height of the 
balancing column of water must be 13*59 times 30 inches, or 
33*975 feet, and the height of the balancing column of air must 
be 773*29 times this, or 26271*52775 feet. In point of fact, the 
height will be a little more than this, as mercury is 13*59593 
times heavier than water, whereas, for simplicity, it has been 
taken here at only 13*59 times heavier. 

EQUATIONS. 

When one quantity is set down as equal to another quantity 
with the sign of equality ( = ) between the two, the whole ex- 
pression is termed an equation. Thus 1 lb. = 16 oz. is an equa- 
tion; and if we represent lbs. by the letter a, and oz. by tho 
letter b, then we shall have the equation in the form 1a or 
a. = 16b. It is clear that the equality subsisting in such an ex- 



NATURE AND USES OF EQUATIONS. 75 

pressiou will not be extinguished by any amount of addition, 
subtraction, multiplication, division, or other arithmetical pro- 
cess to which it may be subjected, provided it be simultaneously 
applied to both sides of the equation — -just as the equality of 
weight shown by a pair of scales between 1 lb. and 16 oz. will 
not be altered if we add an ounce, or pound, or any other weight 
to each scale, or subtract an ounce, or pound, or any other 
weight from each scale. If we add an ounce to each scale, then 
we shall have the equation a + b= 16b + b, or if we subtract 
an ounce from each scale, the equation becomes a — b = 16b — b, 
both of which expressions are obviously just as correct as the 
first one. We may, consequently, add any quantity to each side 
of an equation, or subtract any quantity from it without altering 
the value of the expression. 

If we have such an expression as a — b = 16b — b, and wish 
thereby to know the value of a, we shall ascertain it by adding 
the quantity b to each side of the equation, which will then be- 
come a — b + b = 16b — b+b. Now a— b -f b is obviously equal 
to a, for the value of any quantity is not changed by first sub- 
tracting and then adding any given quantity to it. So likewise 
16b — b + b is obviously equal to 16b, as the — b and + b de- 
stroy one another. The equation thus cleared of redundant 
figures becomes a = 16b. as at first. 

If now we divide both sides of the equation by any number, 
or mutiply both sides \)j any number, we shall find the value of 
the expression to remain without change. For example, if we 

divide by 16 we shall get — = b, or if we multiply by 2 we shall 

get 2a = 32b. Both of these expressions are obviously as true 
as the first one, as they amount to saying that ^th of a pound is 
equal to an ounce, and that 2 lbs. are equal to 32 oz. 

If we have such an expression as a + d = c, and wish to know 
the value of a, we subtract i from both sides of the equation, 
which we have seen we can do without error, whatever quan- 
tity & may be supposed to represent. Performing this subtrac- 
tion we get a + 5— &, or a = c—b ;• and if we know the values 
of c and ft, we at once get the value of a. If we know the 



76 ARITHMETIC OP THE STEAM-ENGINE. 

values of a and c, and wish to find the value of 5, we shalv 
ascertain it by substracting a from each side of the equation, 
which will then become b = c — a. In both of these subtrac- 
tions we may see that we have merely shifted a letter from one 
side of the equation to the other, at the same time changing its 
sign ; and we hence deduce this general law applicable to all 
equations, that we may without error transfer any quantity from 

the one side to the other, if we at the same time change its sign. 

x 
If we have the equation a = t, and if we know the values of 

a and b, but not of #, then, to find the value of x, we multiply 

both sides of the equation by 5, which reduces the equation to 

the form db === x. If, then, a = 2 and 5 = 4, it is clear that 

x = 8. It may be here remarked that db is the same as a x 5, 

and which is quite a different expression from a + b, the one 

meaning a multiplied by 5, and the other a added to b. So like- 

ab _ db 

wise ~r = a and — = b. 

b a 

The utility of such equations in engineering computations is 
very great, not merely as simplifying arithmetical processes, but 
as presenting compendious expressions of important laws, both 
easily remembered and easily recorded. Thus it is found that 
in steam-vessels the power necessary to be put into them, to 
achieve any given speed with any given form of vessel, and any 
given area of immersed midship section, varies as the cube of 
the speed required. If we represent the indicated power by p, 
the speed in knots per hour by s, the area in square feet, and 
the cross section below the water line by a, and if by o we de- 
note a certain multiplier or coefficient, the value of which varies 
with the form of the vessel, but is constant in the same species 

S 3 A 

of vessel, then p = — is an equation which expresses these re- 
lations, and -w e can find the value of p from this equation if we 
know the value of the other quantities, or we can find the value 
of s, or of a, or of c, if we know the values of the other quan- 
tities in the equation. Thus if we multiply both sides of the 
equation by c, we get po = s 3 a, and if we now divide by p we 



EXAMPLES OF EQUATIONS. 77 

get c = — - So also if we divide the equation pc = s 3 a by s 3 , 

PO 

we get the value of a, as we shall then have — j = a ; or if Are 

S 

PC 

divide by a vre get — = s 3 , and taking out the cube root of both 

° A ' ° 

3 / 

/ P' 1 

sides we get A./ — = s. If, therefore, we know the indicator 

' A 

power of a steamer, the immersed area of midship cross section, 
and the coefficient proper for the order of vessel to which the 
particular vessel under examination belongs, we can easily tell 
what the speed will be, as we have only to multiply the indi- 
cator power in horses by the coefficient, and divide by the sec- 
tional area in square feet, and finally to extract the cube root of 
the quotient, which will give the speed in knots per hour. The 
coefficients of different vessels have been ascertained by experi- 
ment. The following are the coefficients of some of the screw- 
vessels of the navy : — 

'Shannon,' 550; 'Simoom,' 500; * Windsor Castle,' 493; 
'Penguin,' 648; 'Plover,' 670; 'Curacoa,' 677; 'Himalaya,' 
695; 'Warrior,' 824; ' Black Prince,' 674. The coefficient of 
the Eoyal Yacht 'Fairy' is 464, and the original coefficient of 
the'Eattler' was 676; but the performance has latterly fallen 
off, and is not now above 500, or thereabout. The original co- 
efficient of the 'Frankfort,' a merchant screw steamer, was 792, 
which was about the best performance at that time attained. 
The larger the coefficient the better is the performance. 



CHAPTER IL 

MECHANICAL PRINCIPLES 01 THE S1EAM-ENGJKB. 



LAW J X>NSERVATION OF FORCE. 

The fundamental principle of Alechanics, as of Chemistry, 
I !. biology, and every department of physical science, is that a 
: : 3 once in being can never cerise to exist, except by its trans- 
formation into some other equivalent force, which, however, 
does not involve the annihilation of the force, as it continue 3 : : 
2x1st in another form. This principle, usually termed the con- 
^ion of force, and sometimes the conservation tf 7?/, is 
only now beginning to receive that wide and distinct recognition 
which its import:..:^ : r lemands; and it will be found that the 
clear apprc-- ansion of t~_:s pervading principle will greatly -::_- 
plify and aid all our in- ssfag a rions in natural science. One very 
obvious inference from the principle is that we cannot manufac- 
ture force out of nou-ix_.\ any more than - - san manufacture 
time, or space, or matter ; and in the various machines for the 
production of power— such its the steam-engine, the wind or 
water mill, or the electro-— u&ve machine — we merely develop 
or liberate the power pent up in the material which we consume 
:: rtnerate the power ; josf as in Eating a clock in motion, we 
liberate the power pent up in the spring. Coal is virtually a 
spring that has been wound up by the hand of nature ; and in 
using it in an engine we are only permitting it to uncoil — im- 



LAW OF VIRTUAL VELOCITIES. 79 

parting thereby to some other agent an amount of power equal 
to that which the coal itself loses. The natural agent employed 
in winding up the springs which our artificial machines uncoil 
is the sun, which by its action on vegetation decomposes the 
carbonic acid which combustion produces, and uses the carbon 
to build up again the structure of trees and plants, that, by their 
subsequent combustion, will generate power; and as coal is only 
the fossil vegetation of an early epoch, we are now using in our 
engines the power which the sun gave out ages ago. So in 
windmills and waterwheels, it is the sun that, by rarefying some 
parts of the atmosphere more than others, causes the wind to 
blow that impels windmills, and the vapours to exhale, which, 
being afterwards precipitated as rain, form the rivers that impel 
waterwheels. In performing these operations the sun must 
lose as much power, in the shape of heat or otherwise, as it im- 
parts; and one of two consequences must ensue — either that 
the sun is gradually burning out, or that it is receiving back in 
some other shape the equivalent of the power that it parts with. 

LAW OF VIRTUAL VELOCITIES. 

One branch of the principle of conservation of force is well 
known in mechanics as the principle of virtual velocities. This 
principle teaches that, as the power exerted in a given time by a 
machine, such as a steam-engine or waterwheel, is a definite 
quantity, and as power is not mere pressure or mere motion, but 
the product of pressure and motion together, so in any part of 
the machine that is moving slowly, the pressure will be great, 
and in any part of the machine moving rapidly, the pressure 
must be small, seeing that under no other circumstances could 
the product of the pressure and velocity — which represents or 
constitutes the power — be a constant quantity. A horse power 
is a dynamical unit, or a unit of force, which is represented by 
33,000 lbs. raised one foot high in a minute of time; and this 
unit is usually called an actual horse power to distinguish it 
from the nominal or commercial horse power, which is merely 
an expression for the diameter of cylinder and length of stroke, 



80 MECHANICS OF THE STEAM-ENGINE 

or a measure of the dimensions of an engine without any refer- 
ence to the amount of power actually exerted by it. If we sup- 
pose that an engine makes one double stroke of 5 feet in the 
minute — which is equal to a space of 10 feet in the minute that 
the piston must pass through, since it has to travel both upward 
and downward — and that this engine when at work exerts one 
horse power, it is easy to tell what pressure must be exerted on 
the piston in order that this power may be exactly attained ; for 
it must be the 10th of 33,000 or 3,300 lbs. ; since 3,300 lbs. mul- 
tiplied by 10 feet is equivalent to 33,000 lbs. multiplied by 1 
foot. Such an engine, if making 10 strokes in the minute, would 
exert 10 horses' power ; if making 20 strokes in the minute would 
exert 20 horses' power; if making 30 strokes in the minute 
would exert 30 horses' power; and in general the pressure on 
the piston in lbs. multipled by the space passed through by the 
piston in feet per minute, and divided by 33,000, will give the 
number of horses' power exerted by the engine. 

It will be clear from these considerations that the circum- 
stance which determines the power exerted by any engine during 
each stroke is — with any uniform pressure of steam — the capacity 
of the cylinder. A tall and narrow cylinder will generate as 
much power each stroke, and will consume as much steam, as a 
short and broad one, if the capacities of the two are the same. 
But the strain to which the piston-rod, the working-beam, and 
the other parts are subjected, will be greatest in the case of the 
short cylinder, since the weight or pressure on the piston must 
be greatest in that case in order to develop the same amount of 
power. Since, too, in the case of an engine exerting a given 
power, the quantity of power is a constant quantity, which may 
be represented by a small pressure acting through a great space, 
or a great pressure acting through a small space, so long as the 
product of the space and pressure remain invariable, it follows 
that in any part of an engine through which the strain is trans- 
mitted, and of which the motion is very slow, the pressure and 
strength must be great in the proportion of the slowness, since 
the pressure multiplied by the motion, at any other part of the 
engine, must always be equal to the pressure multiplied by the 



LAW OF VIRTUAL VELOCITIES. 81 

motion of the piston. In the case of any part of an engine, 
therefore, or in the case of any part of any machine whatever, it 
is easy to tell what the strain exerted will be when we know the 
relative motions of the piston, or other source of power, and of 
the part the strain on which we wish to ascertain, since, if the 
motion of such part be only % of that of the moving force, the 
strain will be twice greater upon that part than upon the part 
where the force is first applied. If the motion of the part be ^ 
of that of the moving force, the strain upon it will be 3 times 
greater than that due to the direct application of the moving 
force ; if the motion be £, the strain will be 4 times greater ; if 4, 
it will be 5 times greater ; if T V, it will be 10 times greater ; if 
f —, it will be 100 times greater : and if any motion of the prime 
mover imparts no appreciable motion to some other part of the 
machine, the strain becomes infinite, or would become so only 
for the yielding and springing of the parts of the machine. "We 
have an example of a strain of this kind in the Stanhope printing 
press, or in the elbow-jointed lever, which consists of two bars 
jointed to one another like the halves of a two-foot rule. If we 
suppose these two portions to be opened until they are nearly 
but not quite in the same straight line, and if they are then in- 
terposed between two planes, and are forced sideways so as to 
bring them into the same straight line, the force with which the 
planes will be pressed apart will be proportional to the relative 
motions of the hand which presses the elbow-joint straight, and 
the distance through which the planes are thereby separated. 
As it will be found that this distance is very small indeed, rela- 
tively with the motion of the hand, when the two portions of 
the lever come nearly into the same straight line, and ceases 
altogether when they are in the same straight line, so the pressure 
acting in separating the planes will be very great indeed when 
the parts of the lever come into nearly a straight line, and is in- 
finite when they come really into a straight line ; or it would be 
so but for the compressibility of the metal and the yielding of the 
parts of the apparatus. 

It is perfectly easy, with the aid of the law of virtual veloci- 
ties, to determine the strains existing at any part of a machine, 
4* 



82 MECHANICS OF THE STEAM-ENGINE. 

and also the weight which the exertion of any given force at the 
handle of a crane, winch, screw, hydraulic press, differential 
screw, "blocks and tackle, or any other machine will lift ; for we 
have only to determine the first and last velocities, and in the 
proportion in which the last velocity is slow, the weight lifted 
will be great. Thns, suppose we have a crane, moved by a han- 
dle which has a radius of 2 feet, which turns a pinion of 8 inches 
diameter gearing into a wheel of 4 feet diameter, on which there 
is a barrel of 1 foot diameter for winding the chain upon, it is 
easy to tell what weight — excluding friction — will be balanced 
or lifted by, say a force of 30 lbs. applied at the handle. The 
handle, it is clear, will describe a circle of 4 feet diameter, while 
the pinion describes only a circle of 6 inches diameter, which 
gives us a relative velocity of 8 to 1 ; or, in other words, the 
strain exerted at the circumference of the pinion will be 8 times 
greater than the strain of 30 lbs. applied at the end of the handle ; 
so that it will be 240 lbs. Now the strain of the pinion is im- 
parted to the circumference of the wheel with which it gears ; 
and the strain of 240 lbs. at the circumference of a wheel of 4 
feet diameter will be 4 times greater at the circumference of a 
barrel of 1 foot diameter, placed on the same shaft as the wheel, 
and revolving with it. The weight on the barrel, therefore, 
which will balance 30 lbs. on the handle, will be 4 times 240 lbs., 
or 960 lbs., but for every foot through which the weight of 
960 lbs. is raised, the handle must move through 32 feet, since 
30 lbs. moved through 32 feet is equivalent to 960 lbs. moved 
through 1 foot. So also in the case of a screw press, the screw 
of which has a pitch of say half an inch, and which is turned 
round by a lever say 3 feet long, pressed with a weight of 30 lbs. 
on the end of it, we have here a moving force- acting in a circle 
of 6 feet diameter ; and as at each revolution of the screw it is 
moved downward through a distance equal to the pitch, which 
is |- inch, we have the relative velocities of ^ inch, and the cir- 
cumference of a circle 6 feet in diameter. Now the proportion 
of the diameter of a circle to its circumference being 1 to 
3'1416, the circumference of a circle 6 feet diameter will be 
18*8i96 feet, or say 18 - 85 feet, which, multipled by 12 to reduce 



MODE OF COMPUTING STRAINS. 83 

it to inches, since the pitch is expressed in inches, gives us 226*2 
inches, and the relative velocities, therefore, are 220*2 to -J, or 
452 - 4 to 1. It follows, consequently, that a pressure of 30 lbs. 
applied at the end of the lever employed to turn such a screw as 
has been here supposed, will produce at the point of the screw a 
pressure of 452'4 times 30, or 13,572 lbs., which is a little over 6 
tons. "Whatever the species of mechanism may be — whether a 
hydraulic press, a lever, ropes and pulleys, differential wheels, 
screws, or pulleys, or any other machine or apparatus, this in- 
variable law holds, that with any given pressure or strain at the 
point where the motion begins, the pressure or strain exerted at 
any part of the machine will be in the inverse proportion of its 
velocity — the stress or pressure on any part being great, just in 
the proportion in which its motion is slow. 

In the case of a lever like the beam of a pair of scales, which 
has its fulcrum in the middle of its length, the application of 
any force or pressure at one end of the beam will produce an 
equal force or pressure at the other end ; and both of the ends 
will also move through the same distance if motion be given to 
either. But if the fulcrum, instead of being placed in the mid- 
dle of the beam, be placed intermediately between the middle 
and one end, we shall then have a lever of which the long end 
is 3 times the length of the short one, and a pound weight 
placed at the extremity of the long end, will balance 3 lbs. 
weight placed at the extremity of the short end. If, however, 
the short end be moved through 1 foot, the long end will be 
simultaneously moved through 3 feet; and 3 lbs. gravitating 
through 1 foot expresses just the same amount of mechanical 
power as 1 lb. gravitating through 3 feet. In a safety-valve, 
pressed down by a lever 5 feet long, while the point which 
presses on the spindle of the safety-valve is 6 inches distant 
from the fulcrum, we have a lever, the ends of which have a 
proportion of.^-to 5, or 1 to 10 ; so that every pound weight 
hung at the extremity of the long end of such a lever, will be 
equivalent to a weight of 10 lbs. placed on the top of the valve 
itself. In the case of a set of blocks and tackle, say with 3 
Bheaves in each block, and, therefore, with 6 ropes passing from 



84 MECHANICS OF THE STEAM-ENGINE. 

one block to the other, it is clear that if the weight to he lifted 
be raised a foot, each of the ropes will have been shortened a 
foot, to do which — as there are 6 ropes — the rope to which the 
motive power is applied must have been pulled out 6 feet. "We 
have, here, therefore, a proportion of 6 to 1 ; or, in other words, 
a weight of 1 cwt. applied to the rope which is pulled, would 
balance 6 cwt. suspended from the blocks. 

It is a common practice among sailors in tightening ropes — 
after having first drawn the rope as far as they can by pulling it 
towards them — to pass the end of the rope over some pin or 
other object, and then to pull it sideways in the manner a harp 
string is pulled, taking in the slack as they again release it. 
This action is that of the elbow-jointed lever reversed; and inas- 
much as the tightened rope may be pulled to a considerable dis- 
tance sideways, without any appreciable change in its total 
length, the strain imparted by this side pulling is great in the 
proportion of the smallness of the distance through which any 
given amount of side deflection will draw the rope on end. 

A hydraulic press is a machine consisting of a cylinder fitted 
with a piston, beneath which piston water is forced by a small 
pump ; and at each stroke of the pump the piston or ram of the 
hydraulic cylinder is raised through a small space, which will 
be equal to the capacity of the pump spread over the area of 
the hydraulic piston. If, for example, the pump has an area of 
1 square inch, and a stroke of 12 inches, its capacity or content 
will be 12 cubic inches ; and if the piston has an area of 144 
square inches, it is clear that the pump must empty itself 12 
times to project 144 cubic inches of water into the cylinder, and 
which would raise the piston or ram 1 inch. In other words, 
the plunger of the pump must pass through 12 times 12 inches, 
or 144 inches, to raise the piston of the hydraulic cylinder 1 
inch, so that the motion of the piston or ram of the hydraulic 
cylinder being 144 times slower than that of the plunger of the 
pump, it will exert 144 times the pressure that is exerted on the 
piston of the pump to move it. "When, therefore, we know the 
amount of pressure that is applied to move the plunger of the 
p ;mp, wg can easily tell the weight that the hydraulic piston 



MODE OF COMPUTING STRAINS. 85 

will lift, or the pressure that it will exert ; and, indeed, this 
pressure will be greater than that on the pump in the proportion 
of the greater area of the hydraulic piston, relatively with that 
of the pump plunger, and which in the case supposed is 144 to 1. 

There are various forms of differential apparatus for raising 
weights, or imparting pressure, in which the terminal motion is 
rendered very slow, and therefore the terminal pressure very 
great, by providing that it shall be the difference of two mo- 
tions, very nearly equal, but acting in opposite directions. 
Thus, if the bight of a rope be made to hang between two 
drums or barrels on which the different ends of the rope are 
wound, and one of which barrels pays the rope out, while the 
other winds it up at a slightly greater velocity than that with 
which it is unwound by the other, the bight of the rope will be 
very slowly tightened ; and any weight hung upon the bight 
will be lifted up with a correspondingly great force. Then there 
are forms of the screw press in which the screw winds itself up 
a certain distance at one end, and unwinds itself nearly the 
same distance at the other end ; so that, at each revolution, it 
advances the object it presses upon through a distance equal to 
the difference of the winding and unwinding pitches ; and as 
this difference may be made as small as we please, so the pres- 
sure may be made as great as we please. The effect of using 
these differential screws is the same as would be obtained if we 
were to use a single common screw having a pitch equal to the 
differences of the pitches. But in practice such a pitch would 
be too fine to have the necessary strength to resist the pressure ; 
and consequently differential screws are in every respect prefer- 
able. 

It is easy to tell what the pressure exerted by a differential 
screw will be, when we know the actual advance it makes at 
each revolution. Thus, suppose the pitch of the unwinding or 
screwing-out part of the screw to be half an inch, or ^%% of an 
inch, and the pitch of the winding or screw r ing-in part of the 
screw to be ■£$£$ of an inch, then the distance between the 
winding and unwinding nuts will be increased -y^o%— iwo 01 " 
TaVoth part of an inch at each revolution. The pressure exerted 



86 MECHANICS Or IHE STEAM-ENGINE. 

by such a screw will consequently be the same as if the pitch 
were TT jVoth part of an inch ; and snch pressure may be easily 
computed in the manner already explained. 

There are various forms of differential gearing employed in 
special cases — not generally for the purpose of generating a 
great pressure, but for the purpose of generating a slow motion 
with few wheels ; though a great pressure is an incident of the 
arrangement, if the terminal motion be resisted. Thus, if we. 
place two bevel wheels on the same shaft, with the teeth facing 
one another, and cause the two wheels to make the same num- 
ber of revolutions in opposite directions, and, further, if we 
place between the two wheels, and on the end of a crank or 
ai-m capable of revolving between them, a bevel pinion, gearing 
with the two wheels, then it will follow — if the two wheels 
have the same number of teeth — that the bevel pinion will 
merely revolve en its axis, but that this axis or crank will be 
itself stationary. If, however, one wheel is made with a tooth 
more than the other wheel, then it will follow that the crank or 
arm carrying the bevel pinion will be advanced through the dis- 
tance o'f one tooth by each revolution of the wheels, and the 
arm will consequently have a very slow motion round the shaft, 
and will impart a correspondingly great pressure to any object 
* by which that motion is resisted. Differential gearing is princi- 
pally employed for drawing along, very slowly, the cutter block 
in boring mills ; and many of its forms are very elegant. It is 
also employed in various kinds of apparatus for recording the 
number of strokes made by an engine in a given time. But the 
same conditions which render the motion slow, also render it 
forcible ; without any reference to the forms of apparatus by 
which the transformation is produced. 

These expositions are probably sufficient to show how the 
pressure exerted by any machine may be computed ; and as the 
pressure is only another name for the strain, we may thence 
discover how to apportion the material to give the necessary 
strength. The very same considerations will enable us to deter- 
mine the strains existing at any part of an engine, or at any 
part of any structure whatever ; and when we know the 



MODE OF COMPUTING SI RAINS. 87 

amount of the strain, it becomes easy to tell how much mate- 
rial, of any determinate strength, we must apply in order to re- 
sist it. Let us suppose, for example, that we wished to know 
the strain which exists at any part of the main beam of a land 
engine, in order that we may determine what quantity of metal 
we should introduce into it to give it the necessary strength. 
How if we suppose the fly wheel to be jammed fast when the 
steam is put on the engine, it is clear that the connecting-rod 
end of the beam will be thereby fixed, and will become a ful- 
crum round which the piston-rod will endeavour to force up the 
beam, lifting the main centre with twice the pressure that the 
piston exerts ; since if we suppose the main centre to be a 
weight, and the fulcrum to be at the end of the beam, this 
weight would only be moved through one inch, when the piston 
moved through 2 inches, so that the lifting pressure upon this 
point would be twice greater than that upon the piston, and the 
main centre must consequently be made strong enough to with- 
stand this strain. If, however, we suppose the main centre to 
be sufficiently strong, we may dismiss all consideration respect- 
ing it, and may consider the beam, which will be thus fixed at 
two points, as a beam projecting from a wall, which an upward 
or downward pressure is applied to break. 

Now in any well-formed engine beam, and indeed in all 
metal beams of proper construction, the strength is collected at 
the edges ; and the web of the beam acts merely in binding 
into one composite mass the areas of metal which are to be 
compressed and extended. The edges of the beam may be in 
fact regarded as pillars, which it is the tendency of the strain 
applied to the beam to crumple up on the one edge, and tear 
asunder on the other edge ; and the whole strength of the beam 
may be supposed to reside in these pillars, since if they were to 
break the rest of the beam would at once give way. The 
strength of any given material to resist compression is not neces- 
sarily, nor always the same as the strength to resist compression. 
In the case of wrought-iron the stretching strength is about 
twice greater than the crumpling strength ; whereas, in the case 
of cast-iron the crushing strength is between 5 and G times 



88 MECHANICS OF THE STEAM-ENGINE. 

greater than the tensile strength. In the ease of an engine 
beam, which has the strain applied alternately in each direction, 
the weakest strength must necessarily be that on which oui 
computations are based ; and in machinery it is not advisable to 
load cast-iron with a greater weight than 2,000 lbs. per square 
inch of section. Now if we suppose, for the sake of simplify- 
ing the computation, that the depth of the beam at the centre 
is equal to its length, then it is clear that if the end of the beam 
moves through any given distance, a point on the edge of the 
beam over or below the main centre will move through the 
same distance, having the same radius ; and if we suppose that 
the depth of the beam is equal to half its length, then a point 
on the edge of the beam, over or below the main centre, will 
move through half the space that the end of the beam moves 
through, and at such point there will consequently be twice the 
amount of strain existing than is exerted upon the piston. For 
every 2,000 lbs., therefore, of pressure on the piston, there 
ought to be strength enough at the edge of the beam to with- 
stand a strain of 4,000 lbs. ; but as this strength has to be di- 
vided between the two edges of the beam, there should be 
strength enough at each end to bear 2,000 lbs. without straining 
the metal more than 2,000 lbs. per square inch of section. In 
other words, with such a proportion of beam there ought to be 
a square inch of section in the top and bottom flanges or mould- 
ings of the beam,' for each 2,000 lbs. pressure or load upon the 
piston. In land engines a common proportion for the depth of 
the beam is the diameter of the cylinder ; and a common pro- 
portion for the length of stroke is twice the diameter of the cyl- 
inder, while the length of the beam is commonly made equal to 
three times the length of the stroke. "With these proportions 
the length of the beam will be equal to six times its depth ; and 
as the edge of the beam, above or below the main centre, will 
in such a beam have only one-sixth of the motion that the end 
of the beam has, the strain at that part divided between the 
two edges of the beam will be six times as great as the stress 
exerted on the piston. For every 2,000 lbs. pressure, therefore, 
on the piston, there must be about three square inches of sec- 



CASES IN WHICH STRAINS ARE INFINITE. 89 

tional area in the upper and lower flanges or mouldings of the 
beam, or six square inches between the two ; while the web of 
the beam is made merely strong enough to keep the upper and 
lower flanges in their proper relative positions. 

It will be obvious from these considerations, that the prin- 
ciple of virtual volocities enables us to compute the amount of 
strain existing at any part of any machine or engine, as we have 
only to suppose the part to be broken, and to see what amount 
of motion the broken part will have relatively with the motion 
of the prime mover, to determine the amount of the strain. "Wo 
can also easily discern, by keeping this principle in view, how it 
comes that, in the case of marine or other engines arranged in 
pairs, with the cranks at right angles with one another, one of 
the engines is so often "broken hy water getting into the cylin- 
der ; and how necessary, therefore, it is that such engines should 
he provided with safety-valves, so enable the water shut within 
the cylinder to escape. For if water gets into one cylinder, and 
if at or near the end of the stroke the slide-valve shuts off the 
communication both with the boiler and with the condenser, as 
is a common state of things, it will follow that the water shut 
within the cylinder, being unable to escape, will resist the de- 
scent of the piston. As, moreover, the crank of one engine is 
vertical, while that of the other is horizontal, and as when ver- 
tical the crank is virtually an elbow-jointed lever, it will follow 
that one engine, with its greatest leverage of crank, is moving 
into the vertical position the crank of the other engine, in which 
position it will act like an elbow-jointed lever, or the lever of a 
Stanhope press, in forcing down the piston on the water, with a 
pressure that is infinite ; and as the water is nearly incompress- 
ible, and as in the absence of escape-valves it cannot get away, 
some part of the engine must necessarily break. The smaller 
the quantity of water shut within the cylinder, so long as it re- 
sists the piston, the greater the breaking pressure will be ; as the 
crank will, in such case, come more nearly into the vertical po- 
sition where the downward thrust that it exerts is greatest; 
whereas, if there be any large volume of water shut within the 
cylinder, the piston will encounter it before the crank comes 



90 MECHANICS OF THE STEAM-ENGINE. 

near the vertical position, and also before the crank of the othei 
engine conies into the horizontal position in which it exerts the 
greatest leverage in turning ronnd the shaft, as it does when the 
engine is at half stroke. In these as in all other cases in which 
we wish to investigate the strain produced in any machine, or in 
any part of any machine, by any given pressure applied in any 
direction, whether oblique or otherwise, we have only to con- 
sider the amount of motion — in the direction in which the strain 
acts — of that particular part which endures the strain or com- 
municates the pressure, relatively with the amount of simulta- 
neous motion in the prime mover. And if the ultimate motion 
be a tenth, a hundredth, or a thousandth part of the original 
motion, so will the strain or pressure exerted by the prime mover 
at the part where the motion is first communicated be multiplied 
ten, a hundred, or a thousand fold. 

NATUEE OF MECHANICAL POWER. 

Mechanical power, or, as it is sometimes defined, worlc, or vis 
viva, is pressure acting through space ; and the law of the con- 
servation of force teaches that power once produced cannot be 
annihilated, though it may be transformed into other forces of 
equivalent value. In all machines a certain proportion of the 
power resident in the prime mover is lost, while the rest is util- 
ised and is rendered available for the performance of those 
labours for which power is required. Thus, in a waterwheel, the 
theoretical value of the fall is that due to a certain weight of 
water gravitating through a certain number of feet in the min- 
ute ; and if we know the height of the fall, and the discharge 
of water in a given time, the theoretical value of such a fall can 
be easily computed. But by no species of hydraulic instrument, 
whether a waterwheel, a turbine, a water-pressure engine, a 
Barker's mill, or any other machine, can the whole of the power 
be abstracted from the fall, and be made available for useful 
purposes. About 80 per cent, of the theoretical power of a 
waterfall is considered to be a very satisfactory result to obtain 
in practice ; and the rest is lost by impact and eddies, and by 






MECHANICAL EQUIVALENT OF HEAT. 91 

the friction of the water and of the machine. In the steam- 
engine the motive force is not gravity, but heat ; and just in the 
same way as power is imparted by water in descending from a 
higher to a lower level, so is power imparted by heat in descend- 
ing from a higher to a lower temperature. These two tempera- 
tures are the the temperature of the boiler, and the temperature 
of the condenser ; and it is clear that if the condenser were to 
be made as hot as the boiler, the motion of the engine would 
cease. And just as in a waterfall there is a certain theoretical 
power due to the quantity of gravitating matter and the differ- 
ence of level, so in a steam-engine there is also a certain theoret- 
ical power due to the quantity of heated matter, and the differ- 
ence of temperature ; but in utilising the power of steam-en- 
gines, this theoretical limit is not approached so nearly as in hy- 
draulic machines. The great fault of the steam-engine is that 
the larger part of the attainable fall is lost. Thus, if we sup- 
pose the temperature of the furnace to be 2,500° Fahrenheit, and 
the temperature of the boiler to be 250°, while that of the con- 
denser is 100°, we utilise pretty effectually the power represented 
T>y the difference in temperature between 100° and 250° ; but 
the difference between 250° and 2,500° is not utilised at all. 
The consequence of this state of things is that not above one- 
tenth of the power theoretically due to the fuel consumed, is 
utilised in the best modern steam-engines — the rest being 
thrown away. 

MECHANICAL EQUIVALENT OF EEAT. 

If the law of the conservation of force be an invariable law 
of nature, we shall naturally expect to find that the power which 
is consumed when a steam-engine or other machine is set to ex- 
ecute useful work, reappears as an equivalent force in somo 
other form. This consequently is the case. "When an engine is 
employed to pump water, we have obviously the equivalent of 
(4ie force in the water pumped to a higher level ; and if this 
water were suffered to flow back again, so as in its descent to 
generate power, we should again have the power we before 



92 3IECHAXTCS OS THE STEAil-EXGDvE. 

spent, with, the deductions due to the imperfections of the appa 
rains anployed. In the Base of an engine, however, which ex- 
pends its power in friction, or in snch work as the propulsion 
of a vessel through the water, the reproduction of the equivalent 
of the power expended is not so easily perceived. But in these 
\ --. also, it has been proved by careful experiment, that the 
law of the conservation of force equally obtains. Friction, 
whether of solids or liquids, produces heat ; and in the case of 
an engine which expends its power on a friction brake, or on 
any other analogous object, an amount of heat will be produced, 
such as, if :: could be used without loss in a perfect engine, 
would exactly reproduce the amount of power expended. In 
: ::se of a vessel propelled through the water, the power is 
mainly consumed in overcoming the friction of the water on the 
bottom, of the . and a part is also expended in moving the 

w fcer to a greater ox less extent; and whatever motion the 
water ; ': es, implies a corresponding loss of power by the en- 
gine, which power is nltim : ay expended in moving the parti- 
: ter upon one another. In such operation heat is pro- 
duced ; which heat, if it could be utilised without loss in an 
engine, would exactly reproduce the power expended. It has 
. ai found by carefal experiment, that if the power developed 
by the descent of a pound weight through 772 feet be expended 
in agitating a pound of water, it will raise the temperature of 
that water 1 = Fahrenheit. The fall of any given quantity of 
water through 772 feet is consequently called the Mechanical 
Equivalent of the heat required to raise the same quantity of 
water one degree in temperature ; since theoretically the two 
values are equivalent, and practically the power will produce 
the heat. But we have not yet any form of apparatus by which 
the heat would produce the power ; and before we can possess 
such, we must have an engine ten times better than the best 
form of steam-engine at present in use. There is every reason 
to believe that there is a definite quantity of mechanical power 
or energy in the universe, the amount of which can neither be 
increased nor diminished, though it may be transformed from 
one shape into another: and ieat, light, electricity, and all 



LAWS OF FALLING BODIES. 93 

chemical and vital phenomena are merely phases, more or less 
complex and disguised, of the same elementary force. 

LAWS OF FALLING BODIES. 

Bodies falling to the earth by gravity are drawn thither by a 
species of attraction — constant in amount — which acts in a man- 
ner similar to that which reveals itself when two bodies in op- 
posite electrical states are brought into proximity. "We do not 
know with any certainty the cause of gravity. But we know 
that it would be quite impossible for one body to act upon 
another without some link to connect the two together ; and the 
most probable supposition is, that as sound is a pulsation of the 
air, caused by pulsations of the sounding body, and as light is a 
pulsation in the ether which fills all space, caused by pulsations 
of the illuminating body, so gravity is a similar pulsation in the 
ether, or a pulsation in another kind of ether, caused by the pul- 
sations of the attracting body. We know by experience that sim- 
ilar pulsations may be generated in a piece of iron by sending an 
electric current through it under certain conditions, and which, 
for the time, transforms the iron into a magnet, which will at- 
tract iron in the same way in which the earth attracts heavy 
bodies : and, in like manner, a piece of amber or of sealing-wax 
may be made to attract straws, pieces of paper, and other light 
substances, by being briskly rubbed. The phenomena of the 
gyroscope seem to show that gravity takes an appreciable time 
to act. If a heavy wheel set on the end of a horizontal shaft, 
which is sustained by two suitable supports, be put into rapid 
rotation, the support nearest the wheel may be taken away 
without the wheel falling down, from which it appears that the 
pulsations which produce gravity may be so confounded together 
by the rapid change in the position of the wheel, and conse- 
quently in the rapid change in the direction of the attracting 
pulses or waves, that the phenomena of gravity are no longer 
exhibited, or what remains of them is manifested in a horizontal 
direction instead of in a vertical — the wheel having shifted into 
or towards that direction before the pulsations have had time 



94 MECHANICS OF THE STEA3I-EXG1XE, 

to be completed. We know from experience that conflicting 
sounds may be made to produce silence, and that conflicting 
lights may be made to produce darkness ; and in like manner, it 
would appear, that a conflict in the pulsations which are the 
cause of gravity may sensibly impair or destroy that gravity. 
It has long been known that sunlight consists of light of those 
different colours which are exhibited in the rainbow, and that 
the phenomena of colours in natural objects is produced by the 
property those objects have of absorbing some rays, and reflect- 
ing others, so that in a red object the whole of the rays except 
the red rays are absorbed — and they are reflected; and in a 
blue object the whole of the rays except the blue rays are ab- 
sorbed — and they are reflected; and, as only the reflected rays 
meet the eye, the objects appear of a red or blue colour. It has 
also long been known that in black objects the whole of the rays 
are absorbed, and none reflected; and in white objects that the 
whole are reflected and none absorbed. But the resources of 
photography also enable us to know that there is a species of 
light which is invisible — which has no colour, and no illuminat- 
ing power, but which reveals its existence by the effect it pro- 
duces on photographic preparations. The use of these photo- 
graphic preparations is consequently equivalent to the acquisition 
of a distinct sense ; and one of the most important problems in 
philosophy is to discover how we may acquire the use of artificia. 
senses, whereby we may more effectually interrogate nature. 
There maybe rays in sunlight, and modes of communication be- 
tween one body and another, of which we have no distinct con- 
ception yet ; but there must be a mode of communication, of 
some kind or other, in every case in which cause and effect are 
known to exist. 

The force of gravity, like the force of light or of sound, varies 
in strength with the extension of the orb of propagation ; or, in 
other words, it diminishes in intensity according to a given law 
with the distance from the earth's surface. Nor is this force 
precisely the same in all parts of the world, as near the equator 
it is partly counteracted by the operation of the centrifugal 
force due to the earth's rotation. But all these disturbing causes 



LAWS OF FALLING BODIES. 95 

are of too little effect to be -worth noticing further m a woik of 
this kind; and for all practical purposes we may reckon the 
force of gravity as uniform in all ages, and at all parts of the 
earth's surface. Now, as power is pressure acting through 
space, a falling body just before it reaches the earth must have 
.1 certain proportion of mechanical power stored up in it which, 
if again used to raise the weight, would carry it up once more to 
its original position. This action we observe in a pendulum. If 
we raise the ball of a pendulum sideways through any given 
elevation, it will accumulate so much power or momentum in its 
descent through the arc in which it swings, as to carry it up to 
the same height on the opposite side of the arc, or at least it will 
do so nearly, and would do so wholly but for the friction of the 
suspending point and of the atmosphere, which will cause some 
slight diminution in the amount of elevation at each successive 
beat. If a hole could be made through the centre of the earth, 
and a ball were suffered to drop down it, the velocity would go 
on accelerating — supposing there were no resisting atmosphere 
— -until the centre of the earth were reached ; and the ball would 
then pursue its course with a velocity gradually diminishing un- 
til it reached the surface at the antipodes, when it would come 
to rest, and return — circulating on for ever from surface to sur- 
face, in a manner similar to that in which a pendulum beats in 
its arc. If we suppose an atmosphere to be introduced into the 
hole or tunnel, then the ball would go on accelerating only until 
the resistance of the atmosphere balanced the weight, after 
which no further acceleration would take place. This is the 
same action that exists when a railway-train or a steam-vessel 
is put into motion by an engine. In each case the train, or 
steamer, continues to accelerate until the resistance of the air 
or of the water balances the propelling force, after which, an 
equipoise being established, no further acceleration takes place. 

The velocity which bodies acquire by falling freely by gravity 
proceeds according to a known law, and it is consequently easy, 
when we know the height from which a body has fallen, to de- 
termine its velocity; or conversely, when we know its velocity, 
we can easily tell from what height it must have descended. 



96 



MECHANICS OF THE STEAM-ENGINE. 



Since, too, power is measurable by the distance through whicn 
a given weight is lifted, or through which it descends, it becomes 
easy to tell when we know the weight and velocity of any body, 
how much power there is stored up in it, since this power will, 
in fact, be represented by the weight multiplied by the height 
through which the body must have fallen to acquire its velocity. 
If the successive additions of velocity which a fallen body 
receives in each second of its fall — namely, 32|- feet — be repre- 
sented by the letter g, then the different relations of the time of 
falling, the ultimate velocity, and the height fallen through, will 
be as follows : — 



MOTION OF A HEAVY BODY FALLISTG IN VACUO. 



Time in seconds 1 



Ultimate velocity 

Height fallen through 

Spaces in each second 



1<7 
1? 
1? 



4(7 
16? 



5g 

25? 



6*7 
86? 

11? 



ig 

49? 
13? 



64? 
15? 



9ff 

8lf 

17? 



10 

10j7 

100? 

19? 



The same relations are shown more in detail in the following table: 



MOTION OF A BODY FALLING IN VACUO. 


Time of falling in 
seconds. 


Height fallen in 
feet. 


Velocity acquired in 
feet per second. 


rest 

X 

4 

I 

4 

1 



1_1_ 

n 

16f 2 - 




8-A- 

24i 
S2i 


H 

if 

2 


*°1 9 2 

64i 


40-^ 
48^ 
56& 
64^ 


3 


81f| 

lOOff 

10112JL 
1^I 192 

144$ 


T2| 

80-!% 

88H 
96£ 


4 
5 
6 
1 
8 
9 


25Vi 
4021 
5*79 

1029£ 
1302| 


128-| 
160| 

193 
2251 
257£ 
289i 



LAWS OF FALLING BODIES. 97 

RULES. 

VELOCITY FROM HEIGHT. 

TO FIND THE VELOCITY ACQTTIEED BY A HEAVY BODY IN FALL- 
ING- THROUGH ANY GIVEN HEIGHT. 

Rule. — Multiply the square root of tJie height in feet through 
which the body has fallen by the constant number 8*021. The 
result will be the velocity in feet per second which the body 
will have attained. 

Example. — Suppose a leaden bullet to be dropped from, a 
height of 400 feet :. with what velocity will it strike the 
ground ? 

Here the square root of 400 is 20, and 20, multiplied by 
8*021=160*42, which is the velocity in feet per second which the 
bullet will have acquired on reaching the ground. 

The same result is attained by multiplying the space fallen 
through in feet by 64*333, and extracting the square root of the 
product, which will be the velocity in feet per second. 

VELOCITY EBOM TIME. 

TO FIND THE VELOCITY IN FEET PER SECOND WHICH A BODY 
WILL ACQUIRE BY FALLING FEEELY DURING ANY GIVEN NUM- 
BER OF SECONDS. 

Rule. — Multiply the number of seconds occupied in falling by 
32*166. The result is the velocity of the body in feet per 
second. 

Example. — Suppose a stone to be dropped from such a height 
that it requires four seconds to reach the ground, what velocity 
"will the stone have acquired at the end of its descent ? 

Here four seconds multiplied by 32*166=128*664, which is 
the velocity in feet per second that the stone will have acquired 
on reaching the ground. 
5 



98 MECHANICS OF THE STEAM-ENGINE. 



HEIGHT PROM VELOCITY. 

TO FIND FEOM TFIE TELOCITY ACQUIEED BY A FALLING BOD"2 
THE HEIGHT FEOM WHICH IT MUST HA YE FALLEN, AXD ALSO 
THE TIME OF THE DESCENT. 

Utile. — Divide the square of the acquired velocity in feet per 
second by 64*333, which will give the height in feet from 
which the body must have fallen ; and divide Che height fallen 
by the constant number 16*083, and extract the square root 
of the quotient, which will be the time of descent in seconds. 

Example. — If a stone dropped from the summit of a tower 
strike the ground with a velocity of 120 feet per second, what 
will be the height of the tower, and what the time occupied 
by the stone in its descent ? 

Here 120 squared=14400 and 144400 divided by 64*33 = 
223-84, which is the height of the tower. Further, 223*84 
divided by the constant number 16*083=13*9, the square root of 
which is 3*72, which will be the time in seconds that the stone 
will have taken to fall 223*84 feet. 



HEIGHT FROM: TTEIE. 

TO FIXD FEOM THE TIME OCCUPIED IN THE DESCENT OF A FALL- 
ING BODY WHAT THE HEIGHT IS FEOM WHICH IT MUST HAVE 
DESCENDED. 

Lule. — Multiply the square of the time occupied in the descent 
in seconds by the constant number 16*083. The product is 
the height in feet from which the body must have fallen. 

Example. — If a stone when suffered to fall into a well strikes 
the surface of the water in four seconds, what is the depth of the 
well to the surface of the water ? 

llere 4 seconds squared=16 seconds, and 16 midtiplied by 
16 , 083=257i feet, which is the depth of the well to the surface 
of the water. 



LAWS OF FALLING BODIES. 99 



TIME FHOM VELOCITY. 

TO FIND THE TIME IN SECONDS DURING WHICH A HEAVY BODY 
MUST HAYE CONTINUED TO PALL TO ATTAIN ANY GIYEN 
VELOCITY. 

Rule. — Divide the velocity in feet 'per second by the constant 
. number 32 '16 6. The quotient is the number of second* 
during which the body must have continued to fall to attain 
its velocity. 

Example. — If a stone in falling has attained a velocity on 
reaching the ground of 128*664 feet per second, how many sec- 
onds must it have occupied in its descent ? 

Here 128*664 divided by 32*166=4, which is the number of 
seconds that the stone must have continued to fall to attain its 
velocity. 

TIME PROM HEIGHT. 

TO FIND THE TIME IN WHICH A HEAVY BODY WILL FALL 
THEOUGH A GIVEN HEIGHT. 

Rule. — Divide the height expressed in feet by the constant num- 
ber 16*083, and extract the square root of the quotient, which 
will give the time in seconds in which the heavy body will 
fall through the given height. 

Example. — Suppose a stone to be let fall from a tower 400 
feet high, in what time will it reach the ground ? 

Here 400 divided by 16*083=24*87, and the square root of 
24*87 is 4*986, or very nearly 5 seconds, which is the time that 
would elapse before the stone reached the ground. 

TO FLND THE NUMBER OF FEET PASSED THEOUGH BY A FALLING 
BODY IN ANY GIVEN SECOND OF ITS DESCENT. 

Rule. — Multiply the number of the second by 32-J- and subtract 
from the product 16 T V The remainder will be the numbet 
of feet passed through in the second given. 



100 MECHANICS OF THE STEAM-ENGINE. 

Example. — To find the number of feet passed through by a 
falling body in the ninth second of its descent. 

Here we have 9 x 32£=289£— 16-^=273-^, which is the 
number of feet passed through in the ninth second of the descent. 

MOTION OF FLUIDS. 

The velocity with which water will flow out of a hole at the 
side or in the bottom of a cistern, will be the same as that which 
a heavy body will acquire in falling from the level of the water 
surface to the level of the orifice, and may easily therefore be 
computed by a reference to the laws of falling bodies. The 
atmosphere exerts a pressure of about 14*7 lbs. per square inch, 
or 2116'4 lbs. per square foot, on all bodies on the earth's sur- 
face ; and if the atmosphere be pumped out of the space beneath 
a piston, while suffered to press on its upper surface, the piston 
will be forced downward in its cylinder with a pressure of 14 - 7 
lbs. on each square inch of the piston's area. In a common 
sucking pump the water is drawn up after the piston, in conse- 
quence of the production of a partial vacuum beneath the piston; 
and the water in the well being subjected to the pressure of the 
atmosphere while the pressure is removed from the water in the 
pump barrel, the water rises in the suction pipe, and would con- 
tinue to do so if the pump were raised further and further up, 
until a column of water had been interposed between the pump- 
barrel and the well sufficiently high to balance the weight of the 
atmosphere. The water will cease to rise any higher after this 
altitude has been attained. 

"When we know the weight of a cubic inch or cubic foot of 
water, it is easy to tell the number of cubic inches or cubic feet 
that must be piled upon one another to produce a weight of 
14*7 lbs. on the square inch or 2116*4 lbs. on the square foot ; 
and it will be found to be 408 cubic inches in the case of the 
cubic inches, or a column 1 inch square and 34 feet high, or 34 
cubic feet in the case of the cubic feet. Mercury being about 
13 '6 times heavier than water, a column of mercury 1 inch 
square and 30 inches high will weigh about 15 lbs. A column 



MOTION AMD WEIGHT OF FLUIDS. 101 

of air high enough to weigh 15 lbs., will he 773*29 times higher 
than a column of water of the same weight — water being 773*29 
times heavier than air at the ordinary barometric density of 29*9 
inches of mercury. In other words, the height of a column of 
air 1 inch square and the same density as that on the earth's 
surface, that will weigh 15 lbs., will be 34x773-29 = 25521-86 
feet, or taking the atmospheric pressure at 14*7 lbs., the height 
will be 26214 feet. The velocity therefore with which water 
will rush into a vacuum, will be equal to that which a heavy 
body will acquire in falling through a height of 34 feet. The 
velocity with which mercury will flow into a vacuum, will be 
equal to that which a heavy body will acquire by falling through 
a height of 2\ feet ; and the velocity with which air will flow 
into a vacuum, will be equal to that which a heavy body will 
acquire by falling through a height of 26214 feet. Now the 
velocity which a heavy body will acquire in falling through 34 
feet will be equal to the square root of 34, which is 5 -8 multi- 
plied by the constant number 8-021 ; or it will be 46*5218 feet 
per second, which consequently will be the velocity with which 
water will flow into a vacuum. The velocity with which mer- 
cury will flow into a vacuum will be 12*83 feet per second, 
for the square root of 2\ is 1*6 nearly, and 1*6 multiplied by 
8*021 = 12*8336. The velocity with which ah* weighing 0-080728 
lbs. per cubic foot will flow into a vacuum will be 1298.5999 
feet per second; for the square root of 26214 is 161*9 nearly, 
which multiplied by 8-021 == 1298*5999 feet per second. The 
density of the air here supposed is the density at the tempera- 
ture of melting ice. At the ordinary atmospheric temperatures 
the density will be somewhat less ; and if the density be taken 
so that the height of the homogeneous atmosphere, as it is 
called, or of that imaginary atmosphere which produces the 
pressure — and which is supposed to be of uniform density 
throughout its depth — is 27,818 feet, then the velocity of the air 
rushing into a vacuum will be a little greater than what it has 
been here reckoned at, or it will be 1388 feet per second. 
These velocities it will be understood are the theoretical veloci- 
ties, which can in no case be exceeded ; but which are fallen 



102 MECHANICS OF THE STEA3I-EXGIXE. 

short of in practice to a greater or less extent, depending on the 
size and form of the orifice throngh which the air enters, and 
other analogous circnm stances. 

The velocity with which steam or any vapour or gas what- 
ever will rush into a vacuum, can easily be determined when 
we know its pressure and density ; for taking into account the 
density, or the weight of one cubic foot, we nave merely to see 
how many of these cubic feet must be piled upon one another 
to produce the given pressure or weight upon the square foot of 
lase; and the velocity will be in every case the same as that 
which a heavy body would acquire in falling through the height 
of the column required to produce the weight. Thus it is found 
that the density of steam of the atmospheric pressure is about 
1700 times less dense than water. Mr. "Watt reckoned that a 
cubic inch of water produced a cubic foot or 1728 cubic inches 
of steam, having the same pressure as the atmosphere ; and if 
the pressure of the atmosphere be equal to the pressure pro- 
duced by 34 feet of water, then, if we reckon steam as 1700 
times less dense than water, it would require 1700 columns of 
steam, each 3-1 feet high, placed on top of one another, to exert 
the same weight or pressure as one column of water 34 feet 
high. Xovr 1700 hundred times 34 is 57800, which therefore is 
the height a column of steam 1700 times less dense than water 
would require to have in order to balance the pressure of the 
atmosphere or of 34 feet of water. The velocity which a body 
would acquire in falling through a height of 57800 feet, is 192G'G 
feet per second; for the square root of 57800 is 240 - 2 nearly, 
and 240-2 multiplied by 8'021 =1926-0442 feet per second, which 
is consequently the velocity with which steam of this pressure 
would rush into a vacuum. The velocity with which steam of 
a greater pressure than that of the atmosphere will rush into a 
vacuum, will not be sensibly greater than that of steam of the 
atmospheric pressure. For as the density of the steam increases 
in nearly the same ratio as its pressure, the column will require 
to be as much lower, by virtue of the increased density, as it 
requires to be higher to give the increased pressure. In other 
words, the height of the theoretical column of steam reauired 



VELOCITY OF STEAM INTO THE ATMOSPHERE. 103 

to produce the pressure, will be nearly the same at all pressures ; 
since a low column of dense steam will produce the same press- 
ure as a high column of rare, and the density and pressure ad- 
vance in nearly the same ratio. It may hence be concluded that 
steam of all pressures will rush into a vacuum with a velocity 
of about 2,000 feet per second, if the vacuum be perfect and the 
How unimpeded. 

If steam, instead of being suffered to escape into a vacuum, 
be made to issue into a vessel containing steam of a lower press- 
ure, the velocity of efflux will be the same as that which a 
heavy body would acquire in falling from the top of the column 
of steam required to produce the greater pressure, to the top of 
a lower column of the same steam adequate to produce the 
lesser pressure. Thus if we have steam with a pressure of two 
atmospheres, flowing into steam with a pressure of one atmos- 
phere, then, inasmuch as the density or weight of the steam 
increases very nearly in the same proportion as its pressure, a 
cubic inch of steam with a pressure of two atmospheres will be 
about twice as heavy as a cubic inch of steam with a pressure 
of one atmosphere. Such steam, therefore, instead of being 
1700 times less dense than water, will be the half of this or only 
850 times less dense than water. A column of this steam, 
therefore, 850 times 34 feet=28900 feet high, will exert a press- 
ure of one atmosphere, or about 15 lbs. on each square inch ; 
and a column of twice this height, or 57800 feet, will exert a 
pressure of two atmospheres or 30 lbs. on each square inch. 
The velocity with which the steam will rush from one vessel to 
the other, will be the same as that which a heavy body would 
acquire in falling from the height of the column of the denser 
steam required to produce the higher pressure to the top of the 
column of the same steam of such height as would produce the 
less pressure ; and as in this case the heights of such columns 
will be 1700 x 34 feet, and 850 x 34 feet, or 57800 and 28900 feet, 
the difference of height will be 28900 feet ; and the velocity of 
efflux from one vessel into the other will be equal to that which 
a heavy body would acquire by falling through a height of 
28900 feet. Now the square root of 28900 is 170; and 170 



104 



MECHANICS OF THE STEAM-EX GEST. 



multiplied by 8-021=1363o7 feet per second, "which is the Te- 
locity -with which steam Trith a pressure of two atmospheres 
would rush into steam with a pressure of one atmosphere. This 
consequently may be reckoned as the velocity with which steam 
of 15 lbs. pressure above the atmosphere would rush into the 
atmosphere. Such velocities at different pressures are exhibited 
ie the following table : — 



VELOCITY OF ETFin OP HIGH-PEESSrEE STEAil ESTO THE 
ATilOSPEEEE. 



Pressure ; 




Pre s sure of Velocity of free 


ater.ni above Uk 


efflux in feet per 


steam at ove the efflux in feet per 


atmosphere. 


5 : '. '. - ~ 


i biosphere, second. 


'.': :. 


fe::. 


I':-:. 


feet 


1 


u _ 


50 


17?1 


o 


663 


60 


1838 


3 


791 


TO 


1577 


4 


SOO 


80 


1919 


5 


ft 


90 


1936 


10 


1241 


!■■: :• 


If '- 


2( 


1504 


110 


1972 


30 




12( 


:" : : 


40 


:-. 


130 


20C i 



This table is computed by taking the difference of the two 
rures for the effective pressure, which effective pressure is 
expressed in pounds per square inch, divided by the weight of a 
cubic foot of the denser fluid in pounds, and the square root of 
the quotient is multiplied by 96. The denser the fluids are the 
less, it is :" e :.:\ will be the velocity of efflux which a given differ- 
eu ;- ■: : | ressnre will create ; for the heights of the columns. : n '. 
also the difference of their heights, will be small in the propor- 
tion of the density of the denser fluid. The more dense the 
fluid is, the larger becomes the mass of matter which a given 
■ ressnre has to move. "With steam of 15 lbs. pressure flowing 
int : steam or air of 15 lbs. pressure, the moving pressure is 1 lb., 
and the velocity of efflux is 482 feet per second. "With steam 
of 101 lbs. pressure fiowin_- int : s:eam or air of 100 lbs. pressure. 



INERTIA AND MOMENTUM. 105 

tlie moving pressure is the same, but the velocity of efflux will 
only be 20T feet per second. 

INERTIA AND MOMENTUM. 

When a body is moved from a state of rest to a state of mo 
Lion, or from a slow motion to a faster, power is absorbed by 
the body ; and when a body is brought from a state of motion 
to rest, or from a fast motion to a slow one, power is liberated 
by the body. The quality which enables a body to resist the 
sudden communication of motion is termed its Inertia ; and the 
quality which enables a body to resist the sudden extinction of 
motion is termed its Momentum. Whatever power a body ab- 
sorbs in being put into motion, it afterwards surrenders in being 
brought to a state of rest ; and the amount of power existing in 
any moving body is measurable by its weight multiplied by the 
square of its velocity, or by the height through which it must 
have fallen by gravity to attain its velocity. 

A railway carriage of ten tons' weight, therefore, moving at 
a speed of 20 miles an hour, will have as great a momentum as 
4 railway carriages weighing 10 tons each moving at the rate of 
10 miles an hour. In like manner the momentum of a cannon 
ball moving at a velocity of 1,700 feet a second, will be 28,900 
times greater than if it moved at a speed of 10 feet per second, 
since the square of 1,700 is to the square of 10 as 28,900 to 1. 
Josephus mentions that some of the battering-rams employed by 
the Eomans in Judea were 90 feet long, and weighed 1,500 tal- 
ents of 114 lbs. to the talent, or 76-3392 tons. The weight of a 
cannon ball which has the same amount of mechanical power 
stored up in it, or which will give the same force of impact 
when moving at a speed of 1,800 feet per second, as the batter- 
ing-ram will do when moving at a velocity of 10 feet per second, 
can easily be determined; for we have only to multiply 76*3392 
tons by the square of 10 and divide by the square of 1,800, 
which will give -0023561 tons, or 5*12776 lbs., as the weight of 
the ball required. 
5* 



206 MECHANICS OF THE STEAM-ENGINE. 

TO FUTD THE QUANTITY OF MECHANICAL POWER EEQUIEED TO COM- 
MUNICATE DIFFERENT YELOCITIES OF MOTION TO HEAYY BODIES, 

Rule. — Multiply the mass of matter oy the height due to the 
velocity it has acquired, supposing that it attained its ve- 
locity oy falling oy gravity. The product is the mechanical 
power communicated in generating that velocity of motion 
in the tody. 

Example 1. — Suppose a waggon on a railway to weigh 2,500 
pounds, what mechanical power must he communicated to it to 
urge it from rest into motion with a velocity of 3 miles an hour, 
or 4*4 feet per second ? 

Now here the height in feet from which a body must have 
fallen to acquire any given velocity will be the square of the 
velocity in feet per second divided by 64-|- ; or it will be the 
square of the quotient obtained by dividing the velocity in feet 
per second by the square root of 64-|, or 8*021. Now 4'4 — 8-021 
=•5487, the square of which is *301 feet, the height that a body 
must fall to acquire a velocity of 3 miles an hour. Hence the 
mechanical power communicated is 2,500 lbs. x '301 ft. = 752*5 
lbs. descending through 1 foot. 

Example 2. — Required the mechanical effect treasured up in 
a cast-iron fly-wheel, the mean diameter of which is 30 feet with 
a sectional area of rim of 60 square inches, and making 20 turns 
in the minute. 

The diameter of the wheel being 30 feet, the circumference 
will be 94*248 feet, and, as the wheel makes 20 revolutions in 
the minute, the velocity of the rim will be 94*248 x 20 =1884*96 
feet per minute, or 31*416 feet per second. Again the cubical 
content of the rim in cubic feet being 60 x 94*248-i-144 = 39*27 
cubic feet, and the weight of a cubic foot of cast-iron being 45 3£ 
lbs., we have 39*27 x 453^=17794*22 lbs. as the weight of the 
rim. Hence the mechanical effect treasured up in the rim of 
(his wheel is 17794*22 x (31*416-^8*021) 2 =268,650 lbs. raised 
one foot high. This it will be observed is about eight actual 
horse-power. The mechanical energy with which the fly-wheel 
of an engine is generally endowed, is equal to the power exerted 



BODIES REVOLVING IN A CIRCLE CENTRIFUGAL FORCE. 107 

in from four to six half strokes of the engine, or two to three 
complete revolutions ; so that the fly-wheel above particularized 
is such as would be suitable for an engine which exerts a power 
of four actual horses, or four times 33,000 pounds raised one 
foot high in each revolution, or 80 horses' power. 

BODIES REVOLVING IN A CIRCLE. 

"When bodies revolve in circles round fixed axes of motion, 
the different particles can have no motion except in circles de- 
scribed round such fixed axes ; and the velocities of the particles 
composing the body must be greater or less, depending upon 
their distance from the centre round which the body revolves. 
To apply the laws of falling bodies to this case we must imagine 
the particles composing such revolving bodies to be divided and 
collected into several small bodies situated at different distances 
from the centre, and therefore moving with different velocities ; 
and then we may determine the power which must be commu- 
nicated to each of the supposed separate bodies to give it the ve- 
locity which it actually possesses. The sum of all the powers 
so determined is the total power which must be communicated 
to the body, to give to it the velocity of motion with which it 
actually revolves. Thus a rod moving about one of its extrem- 
ities may be supposed to be compounded of a number of balls, 
like a string of beads strung on a wire. The velocity of each 
of these balls can then be ascertained, which will enable us to 
compute the mechanical power resident in it, and which will be 
the same as if it moved in a straight line. The sum of the quan- 
tities thus ascertained will be the total mechanical power resi- 
dent in the. revolving body. 

CENTRIFUGAL FORCE. 

The centrifugal force of a body which revolves in any circle 
in a given time, is proportional to the diameter of the circle in 
which it revolves. Thus, in the case of two fly-wheels of the 
same weight but one of twice the diameter of the other, the 



108 MECHANICS OF THE STEAM-ENGINE. 

larger wheel will have twice the amount of centrifugal force 
that the small one has. 

The centrifugal force of a body moving with different veloc- 
ities in the same circle is proportional to the square of the 
velocities with which it moves in that circle ; or, what is the 
same thing, to the square of the number of revolutions per 
formed in a given time. Thus, the fly-wheel of any engine will 
hcve four times the amount of centrifugal force it possessed 
before, if driven at twice the speed. In Mr. "Watt's engines 
with sun and planet wheels, in which the fly-wheel made twice 
the number of revolutions made by the engine, the fly-wheel 
had four times the centrifugal force that would be possessed by 
the same fly-wheel if coupled immediately to the crank. 

The centrifugal force of a body of a given weight, revolving 
with a certain uniform velocity in a circle of a given diameter, 
was investigated by the Marquis de I'Hopital, who gave the rule 
for ascertaining this force that is now generally followed. It is 
founded on the consideration of the height from which the body 
must have fallen by gravity to have acquired the velocity with 
which its centre of gyration moves in the circle which it de- 
scribes. Then as the radius of that circle is to donble the height 
due to the velocity, so is the weight of the body to its centrif- 
ugal force. 

TO FIND THE CENTETFTIGAL FOECE OF A BODY OF A GIVEN WEIGHT 
DEVOLVING IN A CIECLE OF A GIVEN DIAMETEE. 

Rule. — Divide the velocity in feet per second ~by 4*01, and the 
square of the quotient is four times the height in feet due to 
the velocity. Divide this quadrupled height uy the diameter 
of the circle, and the quotient is the centrifugal force when 
the weight of the uody is 1 ; consequently, multiplying it uy 
the weight of the uody gives the actual centrifugal force in 
pounds or tons. 

Example 1- — Suppose that the rim of a fly-wheel 30 feet di- 
ameter and weighing 15718 lbs., moves at the rate of ^7*49 feet 
per second, what will be its centrifugal force? Here we have 



CENTRIFUGAL FORCE OF FLY-WHEELS. 109 

tlie velocity 27*49-s-4*01=6*85, which, squared, is 46*9225 ; and 
this, divided by 30, is 1-564 : so that the centrifugal force is 1*564 
times the weight of the body, or 10*97 tons. 

Example 2. — Suppose that the rim of a fly-wheel which is 20 
feet diameter moves with a velocity of 32£ feet per second : then 
32*1 6-j-4*01=8*02, the square of which is 64*32 feet, which is 
the quadrupled height due to the velocity, and this divided by 
20 feet diameter gives 3*216 times the weight of the rim as the 
centrifugal force. 

Another etjle. — Multiply the square of the number of revolu- 
tions per minute by the diameter of the circle of revolution 
in feet, and divide the product by the constant number 5870; 
the quotient is the centrifugal force of the body in terms of 
its weight, ichich is supposed to be 1. 

Example 1. — Suppose a stone of 2 lbs. weight is placed in a 
sling, and whirled round in a circle of 4 feet diameter, at the 
rate of 120 revolutions per minute : then 120 squared=14400 x 4 
feet diameter=57600-v-5S70=9*81 which is the ratio of the cen- 
trifugal force to the weight ; and, the weight being 2 lbs., the 
centrifugal force acting to break the string and escape is 19*6 lbs. 

Example 2. — In the case of the first fly-wheel 30 feet diam- 
eter, referred to above, we multiply the square of the number 
of revolutions per minute (17|) by the diameter of the circle in 
feet (30), and divide the product by 5870 ; which gives the cen- 
trifugal force in terms of the weight of the body, and 17^ 3 x 30 
-j-5870= 1*564 as before. 

TO FIND THE KATE AT WHICH A BODY MUST EEVOLVE IN ANY CIR- 
CLE, THAT ITS CENTRIFUGAL FORCE MAY BE EQUAL TO ITS 
WEIGHT. 

Rule. — Divide the constant number 5870 by the diameter of the 
circle in feet, and the square root of the quotient is the num- 
ber of revolutions it will make per minute, when the centrif- 
ugal force is equal to the weight. 

Exo.mple. — In a circle of 6*5 feet diameter, a body must re- 
volve about 30 times a minute that its centrifugal force may be 



ilO MECHANICS OP THE STEAM-ENGINE. 

equal to its weight; for 5870-4-6*5=903, the square root of 
which is 30*05 revolutions per minute. 

The mechanical power which must he communicated to a 
solid disc of uniform density, to make it revolve on its axis, is 
the same as that which must be communicated to one-half of its 
weight of matter, to give it motion in a straight line with the 
same velocity with which the circumference of the disc moves 
in a circle. , 

TO DETEEMLXE THE BEESTING STEALS" OF A ELY-WHEEL. 

If we suppose half of a fly-wheel to be securely attached to 
the axis, while the other half is held only by the rim or by bolts 
which it tends to break by its centrifugal force, then there will 
be a velocity at which the centrifugal force of half the rim will 
overcome the cohesion of the metal of the rim, or of the bolts, 
and the wheel will be burst by its centrifugal force. 

in mechanical works it has been usual to reckon the cohesive 
strength of wrought-iron within the limits of elasticity at 17,800 
ibs. per square inch of section, and of cast-iron at 15,300 lbs. 
per square inch of section ; by which is meant that a bar of 
wrought-iron one inch square might be stretched by a weight 
of 17,800 lbs. without injury, and a bar of cast-iron might be 
stretched by a weight of 15,300 lbs. without injury, and though 
somewhat drawn out by such weights, would, like a spiral 
spring, again return to the original length on the weight being 
removed. This estimate for cast-iron is much too high ; and in 
machinery wrought-iron should not be loaded with more than 
4,000 lbs. per square inch of section, and cast-iron should not be 
loaded with more than 2,000 lbs. per square inch of section. 
The breaking tensile strength of good wrought-iron is about 
60,000 lbs. per square inch of section, and of good cast-iron 
about 15,000 lbs. per square inch of section. But both wrought 
and cast-iron will be broken gradually with much less strain 
than would be required to break them at once ; and if the limit 
of elasticity bo exceeded, they will undergo a gradual deteri- 
oration, and will be broken in the course of time. If the velocity 



POWER IN A REVOLVING DISC. Ill 

hi rotation of a cast-iron try- wheel be so great that its centrif- 
ugal force becomes greater than 15,000 lbs. in each square inch 
of the section of the rim, it will necessarily burst, as a wrought- 
iron one would also do if the centrifugal force exceeded 60,000 
lbs. per square inch of section. But to be within the limits of 
safety, a strain of 4,000 lbs. per square inch of section should 
not be exceeded for wrought-iron, and 2,000 lbs. per square inch 
of section for cast. 

TO DETERMINE THE MECHANICAL POWER RESIDENT IN A RE- 
VOLTING DISC. 

Rule. — Multiply one-half of the weight of the revolving disc 
~by the height due to the velocity with which the circum- 
ference of the wheel or disc moves ; the product is the me- 
chanical power communicated. 

Example 1. — Suppose that a grindstone 4*375 feet diameter, 
weighing 3,500 lbs., makes 270 revolutions per minute; what 
power must be communicated to it to give it that motion ? 

The velocity of the circumference will be 61*83 feet per 
second, and the height due to this velocity is 59*4 feet. The 
mechanical power is 1,750 lbs. (half the weight) x 59*4 feet = 
103*950 lbs. raised one foot. 

If the revolving- wheel is not an entire disc or solid circle, but 
only a ring or annulus, it must first be considered as a disc, and 
the effect of the part which is wanting must then be calculated 
and deducted. 

Example 2. — Suppose the rim of a cast-iron fly-wheel to be 
22 feet diameter outside, and 20 feet inside, and that the thick- 
ness of the rim is 6 inches, and that the wheel makes 36 revo- 
lutions per minute, what power must be communicated to the 
rim to give it that motion, the weight of the arms being left out 
of the account? 

A solid wheel 22 feet diameter and 6 inches thick would 
contain 190 cubic feet, from which, if we deduct 157 cubic feet, 
which <would be the capacity of a solid wheel 20 feet diameter 
and 6 inches thick, we have 33 cubic feet as the cubical contents 



112 MECHANICS OF THE STEAM-ENGINE. 

of the annulus. Kow in the case of a solid wheel of 22 feet 
diameter, the velocity of the circumference at 36 revolutions 
per minute would he 41*47 feet per second, the height due to 
which would be 26*8 feet, which multiplied by 95 cubic feet (or 
half the mass) gives 2,546 cubic feet of cast-iron, raised 1 foot for 
the power communicated. Then supposing another solid wheel 
20 feet diameter, we shall find by a like mode of computation 
that the power communicated is equivalent to 1,735 cubic-feet 
of cast-iron raised through 1 foot. This deducted from 2,546 
leaves 811 cubic feet raised through 1 foot as the power resident 
in the annulus ; and if we take the weight of a cubic foot of 
cast-iron in round numbers as 480 lbs., we have 389,280 lbs. 
raised 1 foot, for the mechanical power which must be commu- 
nicated to the rim of the fly-wheel in question to give it a ve- 
locity of 36 revolutions per minute. 

The mechanical power which must be communicated to solid 
discs of different diameters, but of the same thickness and den- 
sity, to make them revolve in the same time, is as the fourth 
powers of their diameters. 

CENTRES OF GYRATION AND PERCUSSION. 

The centre of gyration is a point in bodies which revolve in 
circles in which the momentum, or energy of the moving mass, 
may be supposed to be collected. It is in the same point as the 
centre of percussion of revolving bodies, because a revolving 
body, if suffered to strike another body that is either at rest or 
that moves with a different velocity in the same orbit, will 
neither be deflected to the right nor to the left, but will act just 
as if the whole mass of matter were collected in that point. In 
bodies moving forward in a straight line, the centre of percus- 
sion is in the centre of gravity ; but, in bodies revolving in cir- 
cles, the part of the body most remote from the centre of the 
circle moves with a different velocity from the part nearest to 
the centre of the circle. The centre of percussion, therefore, 
cannot be in the centre of gravity in such a case, but at somo 
point nearer the circumference of the circle ; ard the line traced 



TO FIND THE CENTRE OF GYRATION. 113 

by that point will divide the body into two parts, each having 
the same amount of mechanical power treasured in them, or 
each requiring the same amount of mechanical power to put 
them into revolution at their existing velocity. If the body, 
therefore, could be divided instantly, and without violence, 
through the line traced by the centre of gyration, each portioc 
of the body would continue to revolve with its former velocity. 
The point tracing the line which thus divides the body is the 
centre of percussion, and also the centre of gyration, and in re- 
volving bodies these centres are identical. If a given pressure 
act, through a given space, upon a body at its centre of gyration, 
in the direction of a tangent to the circle which that centre must 
describe round the fixed centre of motion, such an amount of 
power will move the centre of gyration with the same velocity 
in its circle of revolution, as it would move an equal mass of 
matter in a right line by acting at the centre of gravity of the 
mass. If the whole mass of the revolving body could be col- 
lected into its centre of gyration, the mechanical power resident 
in the body would be represented by multiplying the total 
weight of the body by the square of the velocity of the centre 
of gyration. 

TO FIND THE DISTANCE OF THE CENTSE OE GTEATTON OF AIsl 
EEVOLVIXG BODY EEOM THE CENTEE OE AXIS OF MOTION". 

Kule. — Multiply tlie zveight of each particle, or equal small 
portion of the oody, oy the square of its distance from the 
axis, and divide 'the sum of all these products oy the weight 
of the whole mass ; the square root of the quotient will oe 
the distance of the centre of gyration from the axis of motion. 

Example. — Suppose three cannon balls to be fixed on a 
straight rod which is assumed to be without weight ; one ball, 
weighing 2 lbs., is fixed at a distance of 10 inches from the axis 
of motion ; another, which weighs 4 lbs., at 6 inches 1 distance ; 
and the third, which weighs 6 lbs., at 4 inches' distance; then 
the distance of the centre of gyration from the axis of motion 
will be found thus : 10 inches squared =100 ; x 2 lbs. = 200 ; 



114 MECHANICS OF THE STEAM-ENGINE. 

6 inches squared x 4 lbs. = 144 ; and 4 inches squared x 6 lbs 
= 96. The sum of these products is 440, which divided by the 
sum of the weights, or 12 lbs. = 36'66, the square root of which, 
G-05 inches, is the distance of the centre of gyration from the 
axis of motion ; therefore, a single ball of 12 lbs. weight, placed 
at 6*05 inches from the axis of motion, and making the same 
number of revolutions in any given time, would have the same 
amount of mechanical power resident in it as the three balls in 
their several places, as at first supposed. 

The mechanical power which must be communicated to a 
straight uniform rod or lever, to put it in motion, about one of 
its extremities, as a fixed centre or axis, is the same as that 
which must be communicated to an equal weight of matter to 
give it motion in a straight line, with '57,735 of the velocity 
with which the extremity of the lever moves in its circle. The 
point in the revolving lever which moves with that velocity is 
the centre of gyration. 

The mechanical power which must be communicated to a 
solid circular wheel to make it revolve upon its axis, is the same 
as that which must be communicated to an equal weight of 
matter to give it motion in a straight line with *7071 of the 
velocity with which the periphery of the wheel moves within 
its circle, and the point in the radius of the wheel which moves 
with *7071 of the velocity of the circumference is the centre of 
gyration. The weight of the revolving body, multiplied into 
the height due to the velocity with which the centre of gyration 
moves in its circle, in all cases represents the mechanical power 
which must be expended upon the body to give it the velocity 
of rotation that it possesses. 

THE PENDULUM. 

The point from which the pendulum is hung is termed the 
centre of suspension. The effective centre of the ball is an 
imaginary point called the centre of oscillation, and which is so 
situated that the distance from the centre of suspension to the 
centre of oscillation is the same as if the rod of the pendulum 
were destitute of weight, and the whole matter of the ball were 



LAWS OP THE PENDULUM. 115 

collected into the centre of oscillation. The centre of oscilla- 
tion is situated in a line passing between the centre of suspen- 
sion and the centre of gravity. 

The number of vibrations made by pendulums of different 
*engths is inversely as the square roots of their lengths. The 
length of the pendulum which will make one vibration every 
second is somewhat different at different parts of the earth's 
surface, but in the latitude of London its length is variously 
stated at 39*1393 inches and 89-1386 inches. 

TO FIND THE HEIGHT THROUGH WHICH A BODY WILL FALL IS 
THE TIME THAT A PENDULUM MAKES ONE VIBRATION. 

Rule. — Multiply the length of the pendulum by 4*9348 and it 
will give the height. 

Example.— If we take the length of the seconds pendulum 
at 39*1380 in., then 39*1386x4*9348=193*141 in., which is the 
height that a body will fall by gravity in a second. 

to flnd the length of a pendulum which will peefoem a 
given numbee of vibeations in" a minute. 

Eule. — Divide the constant number 375*36 oy the number of 
vibrations to be made per minute, and the square of the quo- 
tient is the length of the pendulum in inches. 

Example. — If the pendulum has to make 60 vibrations per 
minute, then 375-36-s-60=6-256, the square of which is 39*1386. 
The length 39*1393 is probably still more nearly the correct 
leugth of the seconds pendulum in London. 

TO FIND THE NUMBEE OF VIBEATIONS PEE MINUTE WHICH A 
PENDULUM OF A GIVEN LENGTH WILL MAKF". 

Rule. — Multiply the square root of the length of the seconds 
pendulum by the number of vibrations it maTces per minute, 
and, divide the product by the square root of the length of 
the pendulum whose rate of vibration has to be found. The 
quotient is the number of vibrations per minute that the 
pendulum will maJce. 



116 MECHANICS OF THE STEAM-ENGINE. 

Example.— If the length of a pendulum in the latitude of 
London be 28*75 inches, what will be the number of vibrations 
that it will make per minute? 

Here the square root of 39*1393 multiplied by 60, and divided 
by the square root of 28*75=70 vibrations per minute. 

TO FIND THE LENGTH OP A PENDULUM WHICH SHALL MAKE A 
GIVEN NUMBER OF VIBRATIONS IN A GIVEN TIME IN THE 
LATITUDE OF LONDON. 

Rule. — Multiply the square of the number of seconds in the 
given time by the constant number 39*1393, and divide the 
product by the square of the number of vibrations ; the quo- 
tient will be the required length of pendulum in inches. 

Example. — "What must be the length of a pendulum in order 
to give 35 vibrations per minute ? 

The number of seconds in the given time is 60, hence 60 
multiplied by 60 multiplied by 39*1393 gives 140901*48, which 
divided by 1225 (the square of 35) gives 115*021 inches, the 
length of pendulum required. 

TO FIND THE NUMBER OF VIBRATIONS WHICH WILL BE MADE IN 
A GIVEN TIME BY A PENDULUM OF A GIVEN LENGTH. 

Rule. — Multiply the square of the number of seconds in the 
given time by the constant number 39*1393, divide the prod- 
uct by the given length of the pendulum in inches, and the 
square root of the quotient will be the number of vibrations 
in the given time. 

Example. — The length of a pendulum being 64 inches, what 
number of vibrations will it make in 60 seconds? 

In this case the square of 60 multiplied by 39*1393 gives 
140901*48, which being divided by 64 gives 2201*5856, the square 
root of which 46*09 is the number of vibrations required. 

THE GOVERNOR. 

The governor is a centrifugal pendulum : and its proportions 
may be fixed by the same rules which are smployed to deter- 



REVOLVING PENDULUM OR GOVERNOR. 117 

mine the rates of vibration of pendulums. If we suppose a pen- 
dulum, in the act of vibration, to be at the same time pushed 
sideways by a suitable force, it will nevertheless perform its 
vibration in the same period of time ; and if during its return it 
be again pushed sideways in the opposite direction, it will, 
during this double vibration, have pursued a curvilinear course, 
which, if the deflection be sufficient, will be a circle. A pendu- 
nm, therefore, of the same vertical height as the cone described 
by the arms of a governor, will perform a double vibration in the 
same time as the governor performs one revolution. The rules, 
however, according to which governors are usually proportioned 
are as follow : — 

TO DETEEMINE THE PEOPEE HEIGHT OE THE POINT OF SUSPEN- 
SION OF THE BALLS OF A GOVEENOE, ABOVE THE PLANE IN 
WHICH THEY EEYOLYE WHEN MOYING WITH MEAN VELOCITY. 

Rule. — Divide the number 35,225 by the square of the main 
number of revolutions which the governor maTces per minute. 
The quotient is the proper 'vertical height in inches of the 
point of suspension of the balls above the plane in which 
they revolve, when moving with mean velocity. 

Example. — "What is the proper vertical height of the point 
of suspension above the plane of revolution in the case of a gov- 
ernor making 30 revolutions per minute? 

Here 35225-J-900 (the square of 30) = 39*139, which is the 
same height as that of the seconds pendulum. 

If we have already the vertical height, and wish to know the 
proper time of revolution, we must proceed as follows: — 

tO DETEEMINE THE PEOPEE TIME OF EEYOLUTION OF A GOV- 
EENOE OF WHICH THE VEETICAL HEIGHT IS KNOWN. 

Rule. — Multiply the square root of the height by the constant 
fraction 0*31986, and the product will be the proper time of 
revolution in seconds. 

Example.— lo. what time should a governor be made to re- 
solve upon its j?xis when the vertical height of the cone in which 



118 



MECHANICS OF THE STEAM-ENGINE. 



tlie arms are required to revolve when in their mean posi lion is 
39-1393 inches ? Here 6-256 X 0-31986=2 seconds. 



FRICTION. 

When two bodies are rubbed together they generate heat, 
and consume thereby an amount of power which is the mechan- 
ical equivalent of the heat produced. Clean and smooth iron 
drawn over clean and smooth iron without the interposition of 
a film of oil, or other lubricating material, requires about one- 
tenth of the force to move it that is employed to force the sur- 
faces together. In other words, a piece of iron 10 lbs. in weight 
would require a weight of 1 lb. acting on a string passing over a 
pulley to draw the 10 lb. weight along an iron table. But if the 
surfaces are amply lubricated, the friction will only be from 
^th to g—th of the weight. The friction of cast-iron surfaces in 
sandy water is about one-third of the weight. The extent of the 
rubbing surface does not affect the amount of the friction. 

The experiments of General Morin on the friction of various 
bodies without an interposed film of lubricating liquid, but with 
the surfaces wiped clean by a greasy cloth have been summarised 
by Mr. Eankine in the following table : — 

GEXEEAL MOEIX'S EXPEEEtfEXTS OX FEICTIOX. 



1 

2 
3 
4 
5 
6 
7 
8 
9 

r 
n 

12 
13 
14 
15 
16 
17 
18 
19 



SURFACES. 



Wood on wood, dry 

" " soaped 

Metals on oak, dry 

" " wet ,. 

" " soapy 

Metals on elm, dry 

Hemp on oak, dry 

" u wet 

Leather on oak 

Leather on metals, dry 

" " wet 

" " greasv 

" ■ oily.". 

Metals on metals, dry 

" " wet 

Smooth surfaces, occasionally greased . 

" " continually greased. 

" " best results 

Bronze on lignum Yitse, constantly wet 



Angle of 
repose. 



Friction in 

terms of 

<he weight. 



15 



14° to 26i° 
11F to 2° 
26£° to 31° 
13F to 14£° 

11^° 
11£° to 14* 
28° 
1ST 
to i9i° 

29 F 

20' 

13° 

8f-° 
'¥ to 11F 

16fr° 
4° to 4i° 

3° 
II- to 2° 
~3°? 



•25 to 5 
•2 to -04 
•5 to -6 
24 to -26 
•2 

2 to -25 
•53 
•33 

■27 to -3S 

•56 

•36 

•23 

•15 
•15 to -2 

•3 
D7 to -03 

•05 

03 to -036 
•05? 



LAWS OF FRICTION. 119 

The 'Angle of repose,' given in the first column, is the angle 
tfhich a flat surface will make with, the horizon when a weight 
placed upon it just ceases to move by gravity. The column of 
' Friction in terms of the weight ' means the proportion of the 
weight which must be employed to draw the body by a string in 
order to overcome its friction ; and the proportional weight 13 
sometimes called the Co-efficient of Friction. 

In a paper, of which an abstract has appeared in the Comptcs 
Kcndus of the French Academy of Sciences for the 26th of Apiil, 
1858, M. H. Bochet describes a series of experiments which have 
led him to the conclusion, that the friction between a pair of 
surfaces of iron is not, as it has hitherto been believed, absolutely 
independent of the velocity of sliding, but that it diminishes 
slowly as that velocity increases, according to a law expressed 
by the following formula. Let 

R denote the friction ; 

Q, the pressure ; 

v, the velocity of sliding, in metres per second = velocity in 
feet per second x 0*3048 ; 

/, a, y, constant co-efficients ; then 
R _ f+Y<w 
Q 1 + av 

The following are the values of the co-efficients deduced by 
M. Bochet from his experiments, for iron surfaces of wheels and 
skids rubbing longitudinally on iron rails : — 

/,*for dry surfaces, 0'3, 0*25, 0*2 ; for damp surfaces, 0.14. 

a, for wheels sliding on rails, 0*03 ; for skids sliding on rails, 
0*07. 

7, not yet determined, but treated meanwhile as inapprecia- 
bly small. 

The friction of a bearing or machine per revolution, is nearly 
the same at all velocities, the pressure being supposed to be 
uniform ; but as every revolution absorbs a definite quantity of 
power, and generates a corresponding quantity of heat, it will 
be necessary to enlarge, the rubbing surfaces at high velocities, 
both to prevent the wear from being inconveniently rapid, and 
also to enable the bearing to present a larger cooling surface to 



120 MECHANICS OF THE STEAM-ENGINE. 

the atmosphere, so as to disperse the increments of heat whieli 
in the case of high velocities it will rapidly receive. With the 
same object the lubrication should be ample. The oil should 
overflow the bearing, in the same manner as the oil in a carcel 
or moderator lamp overflows the wick to prevent carbonisation ; 
and, to prevent waste, the oil should be returned by an oil 
pmnp so as to maintain a circulation that will both cool and lu- 
bricate the rubbing parts. 

It was fonnd by General Morin in his experiments, that the 
* Friction of Eest ' was considerably more than the ' Friction 
of Motion,' or, in other words, that it took a greater force to 
move a rubbing body from a state of rest than it afterwards 
took to continue the motion, some of the softer bodies being in 
fact slightly indented with the weight. But in determining the 
friction of machinery, it is the Motion of motion alone that has 
to be considered, so that the other need not be here taken into 
account. 

In the case of rubbing surfaces which are amply lubricated, 
the amount of the friction depends more on the nature of the 
lubricant than upon the material of which the rubbing bodies 
are composed ; and hard lubricants, such as tallow, are more 
suited for heavy pressures ; and thin lubricants, such as almond 
oil, are best suited for mechanisms moving with considerable 
velocity, but on which the strain is small. If too heavy a press- 
ure be applied to a bearing, the oil will be forced out and the 
surfaces will heat ; and this will be liable to take place when 
the pressure on the bearing is much more than 800 lbs. per 
square inch on the longitudinal section of the bearing, though 
in practice the pressure is sometimes half as much again, or 
about 1,200 lbs. per square inch in the longitudinal section of 
the bearing, but such bearings will be liable to heat. Thus in a 
marine engine with a cylinder of 74r| inches diameter, the crank 
pin is 9|- inches diameter, and the length of the bearing is 10 
inches, which makes the area of the longitudinal section of the 
bearing 95 square inches. The area of the cylinder is 4,359 
square inches, and if we take the pressure upon the piston — 
including steam and vacuum — at 25 lbs. per square inch, we 



PRESSURE FOR A GIVEN VELOCITY. 121 

shall have a total pressure on the piston of 108,975 lbs., and, 
consequently, this amount of pressure on the crank pin bearing. 
Now 108,975 lbs., the total pressure, divided by 95 square 
inches, the total surface, gives 1,147 lbs. for each square inch 
of the parallelogram which forms the longitudinal section of the 
bearing. In the engines of Messrs. Maudslay, Messrs. Seaward, 
and most of the London engineers, the pressure per square inch 
put upon the crank pin is less. Thus in their 120-horse engines, 
the diameter of the cylinder is 57|- inches, giving an area of 
2,597 square inches, which multiplied by a pressure of 25 lbs. 
per square inch, gives 64,925 lbs. as the total pressure upon the 
piston. The crank pin is 8 inches diameter, and the bearing is 
8|- inches long, giving 68 square inches as the area of the longi- 
tudinal section ; and 64,925 lbs., the total pressure, divided by 
68 square inches, the total area, gives a pressure of 954-77 lbs. 
per square inch of section. This is still in excess of the 800 
lbs. per square inch to which it is expedient to limit the press- 
ure. But the assumed pressure on the piston is rather large 
in the case of these engines, and the actual pressures will be 
found to agree pretty well with the limit of 800 lbs. on each 
square inch of 'the longitudinal section of bearings which it is 
proper to fix as a general rule in the case of engines moving 
slowly. In the case of fast-moving engines, however, the sur- 
face should be greater. The proportion in which the surface 
should vary with the speed is pretty accurately expressed by the 
following rule : — 

TO FIND THE PEESSTJEE PEE SQUAEE INCH THAT MAT BE PPT 
PPON A BEARING MOVING WITH ANT GIVEN VELOCITT. 

Rule. — To the constant number 50 add the velocity of the hear- 
ing in feet 'per minute, and reserve the sum for a divisor. 
Divide the constant number 70,000 by the divisor found as 
above. The quotient will be the number of pounds per square 
inch that may be put upon the bearing. 

Example 1. — An engine with a cylinder 74|- inches diameter, 
ha3 a crank pin 10 inches diameter. At 220 feet of the piston 
6 



122 MECHANICS OF THE STEAM-ENGINE. 

per minute, and with a stroke of 7# feet, the number of revolu- 
tions per minute will be about 15 ; and as the circumference of 
the crank pin will be about 30 inches or 2| feet, the surface of 
the bearing will travel 15 times 2§, or 37£ feet per minute. 
Adding to this the constant number 50, we have 87£, and 70,000 
divided by 87J = 800, which, at this speed, is the proper press- 
ure to put on each square inch of the longitudinal section of 
the bearing. If it is found on trial that this pressure is ex-' 
cceded, the length or diameter of the pin must be increased 

or both. 

Example 2.— An engine with a cylinder 42 inches diameter, 
has a crank pin ^ inches diameter, the circumference of which 
is 26-7 inches or 2*225 feet. When the engine makes 54-8 revo- 
lutions per minute, the surface of the crank pin will move with 
a speed of 54-8 times 2*225 feet per minute, or 121-8 feet 
per minute. Now 50 + 121*8 = 171*8, and 70,000 divided by 
171.8=407*3, which, at this speed of revolution, is the proper 
load to place'upon each square inch of section in the line of the 
axis. The pressure of steam and vacuum in this engine was 40 
lbs. per square inch ; and as the area of a piston 42 inches diam- 
eter is 1385*4 square inches, the pressure urging the piston will 
be 40 times 1385-4 or 55,416 lbs. Now 55,416 divided by 407*3 
is 136, which must be the number of square inches in the longi- 
tudinal section of the bearing in order that there may not be 
more than 407*3 lbs. on each square inch. To obtain this area, 
the bearing must be 16 inches long, since 8£ inches multiplied 
by 16 inches is 136 square inches! At 60 revolutions, the speed 
of the bearing surface per minute is 60 times 2*225 feet or 133*5 
feet. Now 50 + 133-5=183-5, and 70,000 divided by 183-5=377-4, 
which is the proper load in lbs. for each square inch in the lon- 
gitudinal section of the bearing. At 70 revolutions the speed 
of the bearings is 70 times 2*225 feet, or 155*75 feet per minute. 
Now 50 + 155-75=205-75, and 70,000 divided by 205*75=340*2, 
which is the proper load in pounds to put upon each square 
inch of the longitudinal section of the bearing at this speed of 
rotation. 



PRESSURE FOIl A GIVEN VELOCITY. 123 

*0 FIND THE PEOPER TELOCITY FOE THE SUEFACE OF A BEAR- 
ING WHEN THE PRESSURE PER SQUARE INCH ON ITS LONGITU- 
DINAL SECTION IS GIVEN. 

Rule. — Divide the constant number 70,000 by the pressure per 
square inch on the longitudinal section of the hearing. From 
the quotient subtract the constant number 50. The remain- 
der is the proper velocity of the surface of the bearing in 
feet per second. 

Example 1. — What is the proper velocity of the surface of a 
bearing which has the pressure of 800 lbs. on each square inch 
of its longitudinal section ? Here 70,000 divided by 800 =87*5 ; 
from which if we take 50 there will remain 37*5, which is the 
proper velocity of the bearing in feet per second. 

If we take a hypothetical pressure of 1,400 lbs. per square 
inch of section, we get 70,000 divided by 1,400 = 50, and 
50—50=0 ; so that with such a pressure there should be no 
velocity. Even in cases, however, in which there is very little 
motion, such as in the top eyes of the side rods of marine 
engines, it is not advisable to have so great a pressure upon 
the bearing as 1,400 or even 1,200 lbs. per square inch of 
section. 

Example 2. — What is the proper velocity of the bearing of 
an engine which has a pressure upon it of 407'3 lbs. per square 
inch of section? Here 70,000 divided by 407-3=171-8, which 
diminished by 50 is 121*8, which is the proper speed of the sur- 
face of the bearing with this pressure per square inch of sec- 
tion. If the diameter of the bearing be fy inches, its circum- 
ference will be 2-225 feet, and 121-8 divided by 2-225=54-8 rev- 
olutions, which will be the speed of the engine with these data. 
These proportions allow a good margin, which may often be 
availed of in practice, either in driving the engine faster than 
is here indicated, or in putting more pressure upon the bearing. 
But to obviate inconvenient heating and wear, it will be found 
preferable to adhere, as nearly as practicable, to the proportion 
of surface which these rules prescribe. 



124 MECHANICS OP THE STEAM-ENGINE. 



STRENGTH OF MATERIALS. 

The various kinds of strain to which materials are exposed 
in machines and structures may be all resolved into strains of 
extension and strains of compression ; and in investigating the 
strength of materials there are three fixed points, varying in 
every material, to which it is necessary to pay special regard— 
the ultimate or breaking strength, the elastic or proof strength, 
and the safe or working strength. The tensile or breaking 
strength of wrought-iron, is about 60,000 per square inch of 
section, whereas the crushing strength of wrought-iron is about 
27,000 per square inch of section. In steam-engines where the 
parts are alternately compressed and extended, it is not proper 
to load the wrought-iron with more than 4,000 lbs. per square 
inch of section; or the cast-iron with more than 2,000 lbs. per 
square inch of section. But in boilers where the strain is con- 
stantly in one direction, the load of 4,000 lbs. per square inch 
of section may be somewhat exceeded. The elastic strength is 
the strength exhibited by any material without being perma- 
nently altered in form, or crippled ; for as a piece of iron is 
finally broken by being bent backward and forward, so by ap- 
plying undue strains to any material, it will be finally broken 
with a much less strain than would suffice to break it at once. 
The elastic tensile strength of wrought-iron is between one-third 
and one-fourth of its ultimate tensile strength, and to this point 
the material might be proved without injury. But in proving 
boilers, and many other objects, it is not usual to make the 
proving pressure more than twice or three times the working 
pressure, such proof it is considered involving no risk of strain- 
ing the material while it is adequate to the detection of acci- 
dental flaws if such exist. The following table exhibits the te- 
nacity or tensible strength, and the resistance to compression or 
crashing strength of various materials : — 



STRENGTH OF MATERIALS. 



125 



TENSILE AND CRUSHING STRENGTHS OF VARIOUS MATERIALS PER SQUARL INCH 

OF SECTION. 



MATERIAL. 



METALS. 

Wrought-iron bars 

"Wrought-iron plates 

"Wrought-iron hoops (best best) . 

drought-iron wire* 

Cast-iron (average) 

Cast-iron (toughened) 

Steel 

Cast brass 

Gun metal 

Brass wire 

Cast copper 

Copper sheets 

Copper bolts 

Copper wire 

Silver (cast) 

Gold 

Tin (cast) 

Bismuth (cast) 

Zinc 

Antimony 

Lead (sheet) 



Ash... 
Beech 
Birch . 
Box... 
Elm . . 



Fir (red pine). 

Hornbeam 

.Lance-wood . . . 
Lignum Vitas . . 
Locust 

Mahogany 



Oak. 

Pear , 
Teak 



Granite . . . 
Limestone. 
Slate 



Sandstone 



Brick (weak).. 

Brick (strong). 
Brick (fire).... 

Glass 

Mortar 



Tensile strength 
in lbs. per square 

inch ol section. 



60,000 
52.000 "I 
04,000 ! 
70,000 to f 
100,000 J 
16,500 
25,764 
100,000 to 
180,000 
18,000 
36,000 
50,000 
19,000 
30,000 
38,000 
60,000 
40,997 
20,490 

4,736 

3,137 

7,000 

1,062 

3,000 



17,000 

12,000 

15,000 

20,000 

13,000 
j 10,000 to 
1 14,000 

20,000 

23,000 

12,000 

16,000 
i 8,000 to | 
1 16,000 f 
j 10,000 to ) 
j 19,000 j 
9,800 

15,000 



10,000 to 
12,000 



9,500 
50 



Crushing strength 

a lbs. per square 

inch of section. 



27,000 to 37,000 

varies as cube 
of thickness 
nearly. 

100,000 
130,000 

j- 260,000 

10,000 



9,000t 

9,300 

6,400 
10,300 
10,300 

5,375 to I 

6,200 f 

7,300 

9,900 

8,200 

10,000 
u 

12,000 

5,500 to | 

11,000 j 

4,000 to I 

5,000 J 



4,000 to 1 

5,000 f 
550 to ) 
800 [ 

1,100 

1,700 



* Mr. Pole found the German steel wire used for pianofortes to bear as much aa 268,800 lbs. per 
•qu.ire inch. 

t These values are for dry wood. In wet wood the crushing strmpth is only half as groat. 



126 



MECHANICS OF THE STE A3I-ESGISE . 



It will be remarked that there are very large variations in 
ihe amount of the strength recorded in this table ; and there are 
so many varieties in the quality of the materials experimented 
upon that it is hopeless to expect any absolute agreement in the 
results of different experiments. It will be useful, under these 
circumstances, to set down the main results arrived at by a few 
of the principal experimentalists, leaving the reader to select 
such value as he may consider most nearly agrees with the cir- 
cumstances with which he has to deal, The following are the 
strengths of various metals ascertained by Mr. George Sennie, 
in 1817:— 

TKNSmE 9XEKHGTBS OF METALS EY EKH5IB. 




Tearing weight length of bar 
in lbs.' of a bar oae . ™* $ua» 
™ -. *— ». m leet that 

^ " its own weishu 



.steel 

& : Ilsh malleable iron 
English malleable iron. 

-:-iron 

Cast copper 

Yellow brass 

tin 

Cast lead 



134,256 


39,455 


72,064 


19. 7 40 


55.S72 


16,938 


19,096 


6,110 


19,072 


5,003 


17.958 


5.180 


4,136 


1,496 


1,824 


348 



Professor Leslie, in his Natural Philosophy^ thus explains 
the law of the extension of iron by weights : — 

• A bar of soft iron will stretch uniformly by continuing to 
append to it equal weights till it can be loaded with half as 
much as it can bear; beyond that limit, however, its extension 
will become doubled by each addition of the eighth part of the 
disruptive force. Suppose the bar to be an inch square and 
1,000 inches in length; 36,000 lbs. will draw it out 1 inch, but 
45,000 will stretch it 2 inches; 54,000 lbs. 4 inches; 63,000 
S inches: and 72,000 16 inches, where it would finally break,' 
This popular explanation of the law agrees pretty reariy 



TENSILE AND CRUSHING STRENGTHS. 



127 



with the subsequent deductions of Hodgkinson and other 
enquirers. 

The cohesive strength of woods varies still more than that 
of metals in different specimens, and varies even in different 
parts of the same tree. Thus in Barlow's experiments he found 
the cohesive strength of fir to vary from 11,000 to 13,448 lbs. 
per square inch of section; of ash from 15,784 to 17,850 ; oak 
from 8,880 to 12,008; pear from 8,834 to 11,537, and other 
woods in the same proportions. The following fair average 
values may be adopted : — 

TENSILE STEEXGTHS OE WOODS BY BAELOW. 



KIND OF WOOD. 



Teak 

Oak 

Sycamore 

Beech 

Ash 

Elm 

Memel Fir .... 
Christiana Deal 
Larch 



Tearing 
weight in"lbs. 
of a rod one 
inch square. 



12,915 

11,880 

9,630 

12,225 

14,130 

9,720 

9,540 

12,346 

12,240 



Length in feet 

of a rod one 

inch square that 

would break by 

its own weight. 



36,049 
32,900 
35,800 
38,940 
42,080 
39,050 
40,500 
55,500 
42,160 



The crushing strength of wood, as of most other materials, 
is very different from its tensile strength, and is greatly affected 
by its dryness. The following table exhibits the results of the 
experiments made by Mr. Hodgkinson, to ascertain the crush- 
ing strengths of different woods per square inch of section. 
The specimens crushed were short cylinders, 1 inch diameter 
and 2 inches long, flat at the ends. The results given in the 
firs, column are those obtained when the wood was moderately 
dry. Those in the second column were obtained from similar 
spceimens which had been kept two months longer in a wann 
place : — 



128 



MECHANICS OF THE STEAM-ENGINE. 



CPXSHING- STRENGTHS OF WOODS BY HODGKXXSON. 



KIND OF TVOOD. 



Alder 

Ash 

Bay 

Beech 

English Birch 

Cedar 

Bed Deal 

White Deal.. 

Elder 

Elm 

Eir (Spruce). . 
Mahogany . . . 
Oak (Quebec) 
Oak (English) 
Pine (Pitch). . 
Pine (Bed)... 

Poplar 

Plum (Pry).. 

Teak 

Walnut ..... 
Willow 



Crushing strength per 
square inch of section. 



6831 


to 


6960 


8683 


t< 


9363 


7518 


cc 


7518 


7733 


u 


7363 


3297 


u 


6402 


5674 


« 


5863 


5748 


u 


6586 


6781 


[{ 


7293 


7451 


tc 


9973 


. . 


u 


10331 


6499 


u 


6819 


8198 


(( 


8198 


4231 


I( 


5982 


6484 


a 


10058 


6790 


a 


6790 


5395 


u 


7518 


3107 


a 


5124 


8241 


a 


10493 


. . 


u 


12101 


6063 


ct 


7227 


2898 


« 


6128 



The crushing strength of cast-iron is 98,922 lbs., or, say 
100,000 per square inch of section. 

The strength of wooden columns of different lengths and 
diameters to sustain weights has not been conclusively deter- 
mined, and the longer a column is the weaker it is. But, how- 
ever short it may be, the load upon it should not be above one- 
third of the crushing load, as given above. 



LAW OF THE STSEXGTH OF PILLARS. 

The theory of the strength of pillars propounded by Euler is 
that the strength varies as the fourth power of the diameter 
divided by the square of the length ; and the recent investiga- 
tions of Hodgkinson and others show that this doctrine is nearly 
correct. Thus, in the case of hollow cylindrical columns of cast- 
iron, it is found experimentally that the 3*55th power of the in- 
ternal diameter subtracted from the 3*55th power of the externa 



LAW OF STRENGTH OF PILLARS, 129 

diameter, and divided by the l*7th power of the length, will 
give the strength very nearly, In the case of hollow cylindrical 
columns of malleable iron, it is found that the 3 '5 9th power of 
the internal diameter, subtracted from the 3*59th power of the 
external diameter, and divided by the square of the length, will 
represent the strength ; but this rule only holds when the load 
does T?ct exceed 8 or 9 tons per square inch of section. The 
power of plates to resist compression varies as the cube or more 
nearly as the 2 '878th power of their thickness. But this law r only 
holds so long as the pressure applied does not exceed 9 to 12 
tons per square inch of section. If the load is made greater 
than this, the metal is crushed and gives way. It has been found 
experimentally that in malleable iron tubes of the respective 
thicknesses of *525, *272 and *124 inches, the resistances to com- 
pression per square inch of section are 19'17, 14*47, and 7'47tons 
respectively. Moreover, in wrought-iron tubes 1| inches diam- 
eter and |th of an inch thick, the crushing strength is only 6*55 
tons per square inch of section, while in tubes of nearly the same 
length and thickness, but about 6 inches diameter, the crushing 
strength is 16 tons per square inch of section. The strength of 
a pillar fixed at both ends is twice as great as if it were rounded 
at both ends. The crushing strength of a single square cell or 
tube of wrought-iron of large size, with angle-irons at the cor- 
ners, of the construction adopted in tubular bridges, is when the 
thickness of the plate is not less than one-thirtieth of the di- 
ameter of the cell, about 27,000 lbs. per square inch of section: 
and where a number of such cells are grouped together so as to 
prevent deflection, the crushing strength rises to nearly 36,000 lbs. 
per square inch of section, which is also the crushing strength 
of short wrought-iron struts. The length of independent pillars 
should not be more than 25 times the diameter. 

The weight in lbs. which a square post of oak of any length 
will with safety sustain may be determined as follows : — 

TO DETEEMIXE THE PBOPEE LOAD FOE OAK POSTS. 

Rule. — To 4 times the square of the oreadth in inches add 
half the square of the length in feet, and reserve the sum for 
6* 



130 



MECHANICS OF THE STEAM-ENGINE. 



a divisor. Multiply the cube of the breadth in inches 1$ 
3,960 times the length in feet, and divide the product by the 
divisor found as above. The quotient is the weight in lbs. 
which the oaJo post or pillar will with safety sustain. 

JEJxample. — "What weight will a column of oak 6 inches square 
and 12 feet long sustain with safety ? 

Here the breadth of the post is 6 inches, the square of which 
is 36 ; and 4 times 36 is 144. The length being 12 feet, the 
square of the length is 144, half of which is 72 ; and 72 added to 
144 gives 216 for the divisor. The breadth being 6 inches, the 
cube of the breadth is 216, and the length being 12 feet, we get 
12 times 3,960 which is 47,520. Then 216 times 47,520 is 10,- 
264,320, which divided by 216 gives 47,520, which is the weight 
in lbs. that the post will with safety sustain. 
The following table is computed from the rule given above : — 



SCANTLINGS OF SQUAEE POSTS OF OAK. 

With the weights they will support and the extent of surface of flooring 
they will safely sustain, allowing 1 cwt., 1| cwt., or 2 cwts. to the 
superficial foot of flooring, and calculated for a height of 10 feet. 



Note.- 



•These Scantlings -may be safely used up to 12 feet in height; but above 
that a little extra thickness should be allowed. 









Extent of surface of flooring supported. 


Scantlings. 


Wei 


glit. 














1 cwt. per foot. 


1| cwt. per foot. 


2 cwt. per foot. 


Inches. 


Tons. 


Cwts. 


Square feet. 


Square feet. 


Square feet. 


3x3 


5 


10 


110 


821, 


55 


4x4 


9 


18 


198 


148| 


99 


5x5 


14 


14 


294 


220^ 


147 


6x6 


19 


12 


392 


294 


196 


7x7 


24 


12 


492 


369 


246 


8x8 


29 


10 


590 


442-L 


295 


9x9 


34 


8 


688 


516 


344 


10x10 


39 


4 


784 


588 


392 


11x11 


44 





880 


660 


440 


12x12 


48 


16 


9*76 


732 


488 i 


13x13 


53 


10 


1070 


802|: 


535 


14 x 14 


58 


4 


1164 


873 


582 


15x15 


62 


16 


1256 


942 


628 



LAW OF STRENGTH OF PILLARS. 131 

Similar calculations of the dimensions and loads proper for 
rectangular columns of other woods may be determined by a 
reference to their relative crushing strengths given in page 128. 

The formula given by Mr. Hodgkinson for 4etermining the 
breaking weight of square oak posts where the length exceeds 
30 times the thickness is 

W=2452^-. 

where W is the breaking weight in lbs. ; d the side of the square 
base in inches ; and I the length of the post in feet. 

TO DETERMINE THE PROPER LOAD TO BE PLACED PPOIT SOLID 
PILLARS OF CAST-IRON - . 

The load which may be safely placed upon round posts, or 
solid pillars of cast-iron, may be ascertained by the following 
rule : — 

Rule. — To 4 times the square of the diameter of the .solid 
pillar in inches^ add 0'18 times the square of the length of the 
pillar in feet, and reserve the sum for a divisor. Multiply 
the fourth power of diameter of the pillar in inches by the 
constant number 9562 and divide the product by the divisor 
found as above. The quotient is the weight in lbs. which the 
solid cylinder or post of cast-iron will with safety sustain. 

~Mv. LTodgkinson's formula for the breaking strength in tons of 
solid pillars of cast-iron in the case of pillars with rounded ends 
is — 

Strength in tons=14-*9 

and in pillars wuh flat ends — 

d™ 
Strength in tons=44 - 16— - 

c 

where d is the diameter in inches, and I the length in feet. 

The loads in cwts. which may be put upon solid cylinders or 
columns of cast-iron of different diameters and lengths are ex- 
hibited in the following table: — 



132 



MECHANICS OF THE STEAM-ENGINE. 



WEIGHT IN CWTS. SUSTAINABLE "WITH SAFETY BY SOLID CYL- 
INDERS OR COLUMNS OF CAST-IRON OF DIFFERENT DIAM- 
ETERS AND LENGTHS. 



Diameter 




LENGTH OF COLUMN IN FEET. 




of column 
in iaches. 
























6 1 


8 


10 


12 


r 14 


16 




cwts. 


cwts. 


cwts. 


cwts. 


cwts. 


cwts. 


2 


61 


50 


40 


32 


26 


22 


2^r 


105 


91 


77 


65 


55 


47 


3 


163 


145 


128 


111 


97 


84 


H 


232 


214 


191 


172 


156 


135 


4 


310 


288 


266 


242 


220 


198 


4 2 L 


400 


379 


354 


327 


301 


275 


5 


501 


479 


452 


427 


394 


365 


6 


592 


573 


550 


525 


497 


469 


7 


1013 


989 


959 


924 


887 


848 


8 


1315 


1289 


1259 


1224 


1185 


1142 



In hollow pillars nearly the same laws obtain as in solid. 
Thus in the case of hollow pillars, with rounded ends or movable 
ends, like the cast-iron connecting-rod of a steam-engine, the 
formula is — 



Strength in tons =13 



J3« _ ^3-6 

V" 1 



and in the case of hollow pillars, with flat ends — 



Strength in tons=44 # 3- 



V 



where D is the external and d the internal diameter. The 
strength of a pillar with a cross section of the form of a cross 
was found to be only about half as great as that of a cylindrical 
hollow pillar. It was also found that in pillars of the same 
dimensions, but of different materials, taking the strength of 
cast-iron at 1,000, that of wrought-iron was 1,745, cast steel 
2,518, Dantzic oak 108*8, and red deal 78*5. 

Mr. Hodgkinson's rule for the breaking weight of cast-iron 
beams is as follows : — 



STRENGTH OF CAST-IRON BEAMS AND SHAFTS. 133 
STEEXGTH OF CAST-IEON BEAMS. 

Rule. — Hultiply the sectional area of the bottom flange in 
square inches by the depth of the beam in inches, and divide 
the product by the length, between the supports also in inches. 
Then 514 times the quotient icill be the breaMng weight in 
acts. 

STEEXGTH OF SHAFTS. 

44 lbs. acting at a foot radius will twist off the neck of a 
sliaft of lead 1 inch diameter, and the relative strengths of other 
materials, lead being 1, is as follows: — Tin, 1*4; copper 4*3; 
yellow brass, 4*6; gun metal, 5 ; cast-iron, 9; Swedish iron, 9*5 ; 
English iron, 10*1; blistered steel, 16*6; shear steel, 17; and 
cast steel, 19*5. The strength of a shaft increases as the. ^nbe 
of its diameter. 



CHAPTER ILL 
THEORY OP THE STEAM-EXGEVE. 

I're S team-En gjir is s machine for extracting meehanica. 
power front heat through the agency of water. 

Heat is one form of mechanical power, or more properly, a 
given quantity of heat is the equivalent of a determinate amount 
of mechanical power ; and as heat is capable of producing power, 
so contrariwise power is capable of producing heat. The nature 
of the medium upon which the heat acts in the production of 
the power — whether it be water, air, metal or any other sab- 
stance — is immaterial, except in so far as one substance in; 7 s 
more convenient and manageable in practice than another. But 
with any given extremes of temperature, and any given expen- 
diture of heat, the amount of power generated by any given 
quantity of heat will be the same, whatever be the nature of the 
substance on which the heat is made to act in the generation of 
the power. An 1 just in the proportion in which power is gen- 
erated so will the heat disappear. We cannot have both the 
heat and the power; but as the one is transformed into the 
other, so it will follow that the acquisition of the one entails a 
proportionate loss of the other, and this loss cannot possibly be 
prevented. It has been already explained that as in all cases in 
which power is produced in a steam-engine, there must be a dif- 
ference of pressure on the two sides of the piston, or between the 
boiler and the condenser ; so in all cases in which power is pro- 



NATURE AND EFFECTS 01 HEAT. 135 

d need in any species of caloric engine, there must be a difference 
of temperature between the source of heat and the atmosphere 
or refrigerator. The amount of this difference will determine 
the amount of power, up to a certain limit, which a unit of 
heat will generate in any given engine. But as it has been al- 
ready explained that the mechanical equivalent of the heat con- 
sumed in heating 1 lb. of water 1° Fahrenheit would, if utilised 
without loss, raise a weight of 772 lbs. 1 foot high, it will fol- 
low that in no engine whatever can a greater performance be 
obtained than this, whatever difference of temperature we may 
assume between the extremes of heat and cold. A weight of 
772 lbs. raised 1 foot for 1° Fahrenheit is equivalent to a weight 
of 1389*6 lbs. raised 1 foot for 1° Centigrade; and for conven- 
ience the term foot-pound is now very generally employed to de- 
note the dynamical unit, or measure of power, expressed by a 
weight of 1 lb. raised through 1 foot. A horse-power, or as it 
is now commonly termed an actual or indicator horse-power — 
to distinguish it from a nominal horse-power, which is a mere 
measure of capacity — is a dynamical unit expressed by 33,000 lbs. 
raised 1 foot high in a minute ; or it is 550 foot-pounds per sec- 
ond; 33,000 foot-pounds per minute; or 1,980,000 foot-pounds 
per hour. This unit takes into account the rate ofworlc of the 
machine. 

Heat, like light, is believed to be a species of motion, and 
there are three forms of heat of which a work of this nature re- 
quires to take cognisance — Sensible Heat, Latent Heat, and 
Specific Heat. 

Sensible Heat is heat that is sensible to the touch, or measur- 
able by the thermometer. Latent Heat is the heat which a body 
absorbs in changing its state from solid to liquid, and from liquid 
to aeriform, without any rise of temperature, or it is the heat ab- 
sorbed in expansion. And Specific Heat is an expression for the 
relative quantity of heat in a body as compared with that in 
some other standard body of the same temperature. There is a 
constant tendency in hot bodies to cool, or to transfer part of 
their heat to surrounding colder bodies ; and contiguous bodies 
are said to be of equal temperatures when there ceases to be any 



136 THEORY OF THE STEAM-ENGIXE. 

transfer of heat from one to the other. The most pro m inent 
phenomena of heat are Dilatation. Liquefaction, and Vaporisa- 
tion. 

Difference oetvceen temperature and quantity of neat. — It is 
quite clear that two ponnds of boiling water have just twice the 
quantity of heat in them that is contained in one pound of boil- 
ing water. But it does not by any means follow, nor is it the 
case, that two pounds of boiling water at 212° contain twice the 
quantity of heat that is contained in two pounds of water at 
106°. Experiment indeed shows, that when equal quantities of 
water at different temperatures are mixed together, the resulting 
temperature is the mean of the two, so that if a pound of water 
at 200° be mixed with a pound of water at 100°, we hare a re- 
sulting two pounds of water of 150°. But before we could sup- 
pose that a pound of water at 200° has twice the quantity of heat 
in it that is contained in a pound of water at 100°, it would be 
necessary to conclude that water at 0° or zero, has no heat in at 
whatever. This, however, is by no means the case ; and tem- 
peratures much below zero have been experimentally arrived at 
and even naturally occur in northern latitudes. A pound of ice 
at a temperature below zero, rises in temperature by each suc- 
cessive addition of heat, until it attains the temperature of 82", 
when it begins to melt ; and, notwithstanding successive addi- 
tions being made to its heat, its temperature refuses to rise above 
32 ° until liquefaction has been completed. So soon as all the ice 
has been melted, the temperature of the resulting water will 
continue to rise with each successive increment of heat, until the 
temperature of 212° has been attained, when the water will boil, 
and all subsequent additions to the heat will be expended in evap- 
orating the water or in converting it into steam. Although, 
therefore, a pound of water in the form of steam has only the 
same temperature as a pound of boiling water, it has a great deal 
more heat in it, as is shown by the fact that it will heat to a 
given temperature a great many more pounds of cold water than 
a pound of boiling water would do. 

Absolute zero. — The foregoing considerations lead naturally 
to the inquiry whether, although bodies at the zero of Fahren- 



DIFFERENT THERMOMETERS COMPARED. 137 

Iieit's scale arc still possessed of some heat, there may not, never- 
theless, be a point at which there would be no heat whatever, 
and which point therefore constitutes the true and absolute zero. 
Such a point has never been practically arrived at. But the law 
of the elasticity of gases and their expansion by heat, leads to 
the conclusion that there is such a point, and that it is situated 
401*2° Fahrenheit below the zero of Fahrenheit's scale, or in 
other words that it is — 461-2° Fahrenheit, — 274° Centigrade, 
or — 219*2° Eeaumur. Mr. Rankine has shown, that by reckon- 
ing temperatures from this theoretical zero, at which there is sup- 
posed to be no heat and no elasticity, the phenomena dependent 
upon temperature are more readily grouped and more simply ex- 
pressed than would otherwise be possible. 

Fixed Temperatures. — The circumstance of the temperatures 
of liquefaction and ebullition being fixed and constant, enables 
us to obtain certain standard or uniform temperatures, to which 
all others may easily be referred. One of these standard tem- 
peratures is the melting-point of ice, and another is the boiling- 
point of pure water under the average amospheric pressure of 
14*7 lbs. on the square inch, 2116*8 lbs. on the square foot ; or un- 
der the pressure of a vertical column of mercury 29*922 inches 
high, the mercury being at the density proper to the tempera- 
ture of melting ice. 

Thermometers. — Thermometers measure temperatures by the 
dilatation which a certain selected body undergoes from the appli- 
cation of heat. Sometimes the selected body is a solid, such as 
a rod of brass or platinum ; at other times it is a liquid, such as 
mercury or spirits of wine ; and at other times, again, it is a 
gas, such as air or hydrogen. In a perfect gas the elasticity is 
proportionate to the compression, whereas in an imperfect gas, 
such as carbonic acid, which may be condensed into a liquid, the 
rate of elasticity diminishes as the point of condensation is ap- 
proached. Every gas approaches more nearly to the condition 
of a perfect gas the more it is heated and rarefied,- but an abso- 
lutely perfect gas does not exist in nature. Common air, how- 
ever, approaches sufficiently to the condition of a perfect gas, to 
be a just measure of temperatures by its expansion. 



138 THEORY OF THE STEAM-ENGINE. 

Air aad all other gases expand equally with equal increments 
of temperature ; and it is found experimentally that a cubic foot 
of air at the temperature of melting ice, or 32°, will form 1*365 
cubic feet of the same pressure at the temperature of boiling 
water, or 212°. Thermometers, however, are not generally con- 
structed with air as the expanding fluid, except for the measure- 
ment of very high temperatures. The most usual species of ther 
mometer consists of a small glass bulb filled with mercury, and 
in connection with a capillary tube. The bulb is immersed in the 
substance the temperature of which it is desired to ascertain ; 
and the amount of the dilatation is measured by the height to 
which the mercury is forced up the capillary tube. The ther- 
mometer commonly used in this country is Fahrenheit's ther- 
mometer, of which the zero or of the scale is fixed at the 
temperature produced by mixing salt with snow ; and which 
temperature is 32° below the freezing-point of water. The Cen- 
tigrade thermometer is that commonly used on the continent o± 
Europe ; and it is graduated by dividing the distance between the 
point where the mercury stands at the freezing-point of water, 
and the point where it stands at the boiling-point of water, into 
100 equal parts. Of this thermometer the zero is at the freez- 
ing point of water. Another thermometer, called Beaumur's 
thermometer, has its zero also at the freezing-point of water ; 
and the distance between that and the boiling-point of water 
is divided into eighty equal parts. Hence 80° Eeaumur are equal 
to 100° Centigrade, and 180° Fahrenheit. The correspond- 
ing degrees of these thermometers are shown in the following 
table : — 



DILATATION OF SOLIDS. 



139 



CENTIGRADE, REAUMUR'S, AND FAHEENHEIT's THEEMOMETEES. 



■ Cent 


Reau. 


Fahr. 


Cent. 


Reau. 


Fahr. \ 


Cent. 
29 


Reau. 


Fahr. 


Cent. 


Reau. 


Fahr. 


; 300 


so- 


212- 


64 


51-2 


147-2 


23-2 


84-2 


—6 


—4-8 


21-2 


■ D9 


79-2 


210-2 


63 


50-4 


145-4 


28 


22-4 


82-4 


7 


5-6 


19-4 


i 9S 


7S-4 


20S-4 


62 


49-6 


143-6 


27 


21-6 


80-6 


8 


6-4 


17-6 


97 


77-6 


206-6 


61 


48-8 


141-8 


26 


20-8 


7S-8 


9 


7-2 


158 


96 


76-8 


204-S 


60 


48- 


140- 1 


25 


20- 


77- 


10 


8- 


14- 


95 


76- 


203- 


59 


47-2 


138-2 ! 


24 


19-2 


75-2 


11 


8-8 


12-2 


91 


75-2 


201-2 


58 


46-4 


136-4 


23 


13-4 


73-4 


12 


9-6 


10-4 


93 


74-4 


199-4 


57 


45-6 


1346 


22 


17-6 


71-6 


13 


10-4 


8-6 


92 


73-6 


197-6 


56 


44-8 


132-8 


21 


16-8 


69-8 


14 


11-2 


6-8 


91 


72-8 


195-8 


55 


44. 


131- j 


20 


16- 


68- 


15 


12- 


5- 


90 


72- 


194- 


54 


43-2 


129-2 


19 


15-2 


66-2 


16 


12-8 


3-2 


89 


71-2 


192-2 


53 


42-4 


127-4 


18 


14-4 


64-4 


17 


13-6 


1-4 


SS 


70-4 


190-4 


52 


41-6 


125-6 


17 


13-6 


62-6 


18 


14-4 


—0-4 


8T 


69-6 


1SS-6 


51 


40-3 


123-8 


16 


12-8 


00-8 


19 


15-2 


2-2 


86 


63-S 


186-8 


50 


40- 


122- ! 


15 


12- 


59- 


20 


16- 


4- 


85 


6S- 


1S5- 


49 


39 2 


120-2 


14 


11-2 


57-2 


21 


16-8 


5-8 


84 


67-2 


1S3-2 


4S 


3S-4 


118-4 


13 


10-4 


55-4 


22 


17-6 


7-6 


83 


66*4 


181-4 


47 


37-6 


116-6 


12 


9-6 


53-6 


23 


1S-4 


9-4 


82 


65-6 


179-6 


46 


36-8 


1148 


11 


8-8 


51-8 


24 


19-2 


11-2 


81 


64-8 


177-8 


45 


36- 


113- i 


10 


8- 


50- 


25 


20- 


13- 


80 


64- 


176- 


44 


35-2 


111-2 


9 


7-2 


48-2 


26 


20-8 


14-8 


79 


63*2 


174-2 


43 


34-4 


109-4 


8 


6-4 


46-4 


27 


21-6 


16-6 


78 


62-4 


172-4 


42 


33-6 


107-6 


7 


5-6 


44-6 


28 


22-4 


184 


77 


61-6 


170-6 


41 


32-8 


105-8 


6 


4-8 


42-8 


29 


23-2 


20-2 


76 


60-8 


168-8 


40 


32- 


104- i 


5 


4- 


41- 


30 


24- 


22- 


75 


60- 


167- 


39 


31-4 


102-2 


4 


3-2 


39-2 


31 


24-8 


23-S 


74 


59-2 


165-2 


38 


30-2 


100-4 


3 


2-4 


37-4 


32 


25-6 


25-6 


73 


58-4 


1634 


37 


29-6 


98-6 


2 


1-6 


35-6 


33 


26-4 


27-4 


72 


57-6 


161-6 


36 


28-8 


96-8 


1 


0-8 


33-8 


34 


27-2 


29-2 


71 


56-8 


159-8 


35 


23- 


95- I 





o- 


32- 


35 


2S- 


31* 


70 


56- 


158* 


34 


27-2 


93-2 


_1 


—0-8 


30-2 


36 


28-8 


32-8 


69 


55-2 


156-2 


33 


264 


91-4 


2 


1-6 


28-4 


37 


29-6 


34-6 


6S 


54-4 


154-4 


32 


25-6 


89-6 


3 


2-4 


26-6 


33 


30-4 


36-4 


67 


53-6 


152-6 


31 


24-8 


87-8 


4 


3-2 


24-8 . 


39 


31-2 


38-2 


66 


52-8 


150-3 


30 


24- 


86- I 


5 


4- 


23- 


40 


32- 


40- 


65 


25- 


149- 





















"Water, in common with molten cast-iron, molten bismuth, 
and various other fluid substances, the particles of which assume 
a crystalline arrangement during congelation, suffers an increase 
of bulk as the point of congelation is approached, and expands in 
solidifying. But so soon as any of these substances has become 
solid, it then contracts with every diminution of temperature. 
Water in freezing bursts by its expansion any vessel in which it 
may be confined, and ice, being lighter than water, floats upon 
water. So also for a like reason solid cast-iron floats on molten 
cast-iron. The point of maximum density of water is 8 9*1° Fall- 
rejiheit, and between that point and 32° the bulk of water in- 



140 THEORY OF THE STEAM-ENGINE. 

creases by cold. A cubic foot of water at 82 c weighs 62*425 
lbs., whereas a cubic foot of ice at 32° weighs only 57*5 lbs. 
There is consequently a difference of nearly 5 lbs. in each cubic 
foot, between the weight of ice and the weight of water. 



DILATATION. 

Dilatation of Solids. — A solid body of homogeneous texture 
will dilate uniformly throughout its entire bulk by the applica- 
tion of heat. Thus, if it be found that a bar of zinc is increased 
one 340th part of its length by being raised in temperature from 
32° to 212°, its breadth will also be increased one 340th part, 
and its thickness will be increased one 340th part. It is found, 
moreover, that equal increments of heat produce equal augmen- 
tations of volume in nearly all bodies, at all temperatures, until 
the melting-point is approached, when irregularities occur. 
Different solids dilate to different amounts when subjected to 
the same increase of temperature, and advantage is taken of this 
property in the arts in the construction of time-keepers and 
other instruments. Thus, in Harrison's gridiron pendulum, the 
ball is composed of bars of different metals, some of which ex- 
pand more than the others at the same temperature ; and as the 
bars which expand the most are fixed at the lower ends and ex- 
pand upwards, they compensate for the expansion of the pendu- 
lum rod in the opposite direction, and maintain the centre of os- 
cillation in the same place. The following table exhibits the 
rates of dilatation of various solids, as ascertained by the best 
authorities : — 



DILATATION PRODUCED BY HEAT. 



141 



DILATATION OF SOLIDS BY HEAT. 



Bodies. 



Dilatation in Tractions. 



Decimal. Vulgar. 

Dilatation from 32° to 212°, according to Lavoisier and Laplace. 



Flint Glass (English) 

Flatinurn (according to Borda) . 

Glass (French) with lead 

Glass tube without lead 

Ditto 

Ditto 

Ditto 

Glass (St. Gobain)* • • ■ 

Steel (untempered) 

Ditto 

Ditto 



Steel (yellow temper) annealed at 65°. 

Iron, soft forged 

Iron, round wire-drawn 

Gold 

Gold (French standard) annealed 

Gold (ditto) not annealed 

Copper 

Ditto 

Ditto 

Brass. 

Ditto 

Ditto 

Silver (French standard) 

Silver 

Tin, Indian 

Tin, Falmouth 

Lead 



According to Smeaton 



Glass, white (barometer tubes) 
Steel 



Steel (tempered) 

Iron 

Bismuth 

Copper 

Copper 8 parts, tin 1 . 

Brass cast 

Brass 10 parts, tin 1. 



0-00081166 


L_ 

12 32 


0-00085655 


1 
1 J 67 


0-00087199 


TTTT 


0-00087572 


l 
1 1 42 


0-00089694 


1 

1115 


0-00089760 


1 _ 

1114 


0-00091750 


1 

1U90 


0-00089089 


1 1 2~~2 


0-00107880 


1_ 
92 7 


0-00107915 


92 7 


0-00107960 


926 


0-00123956 


1 
807 


0-00122045 


1 
819 


0-00123504 


_1_ 

8 1 


0-00146606 


_1_ 

682 


0-00151361 


_JL_ 

6 6 1 


0-00155155 


1 
645 


0-00171220 


1 
5 84 


0-00171733 


1 
6 82 


0-00172240 


1 

5 8 1 


0-00186670 


1 

5 3 5 


0-00187821 


1 

5 4 2 


0-00188970 


1 
629 


0-00190868 


1 

624 


0-00190974 


62"4 


0-00193765 


_A_ 
5 J 6 


0-00217298 


460 


0-00284836 


1 
35 1 


0-00083333 


1 
12 


0-00108333 


1 
9 2 3 


0-00115000 


1 
8 7 


0-00122500 


1 

8 l~5 


0-00125833 


1 

5 9 5 


0-00139167 


1 

7 1 9 


0-00170000 


1 

688 


0-00181667 


1 
56 


0-00187500 


1 
5 a 3 


0-00190833 


1 
5 24 



U2 



THEORY OF THE STEAM-ENGINE. 



DILATATION OF SOLIDS BY HEAT — continued. 



Dilatation in Fractions. 



Bodies. 

Brass wire 

Telescope speculum metal 

Solder (copper 2 parts, zinc 1). . . . 

Tin (fine) 

Tin (grain) 

Solder white (tin 1 part, lead 2). . . 
Zinc 8 parts, tin 1, slightly forged. 

Lead 

Zinc 

Zinc lenthened -^ by hammering. 
Palladium ( Wollaslon) 



Decimal. 

0-00193333 
0-00193333 
0-00205833 
0-00228333 
0-00248333 
0-00250533 
0-00269167 
0-00286667 
0-00294167 
0-00310833 
0-00100000 



According to Dulong and Petit 



Platinum 



Glass , 



Iron. 



Copper. 



32° 


to 212° 


32° 


to 572° 


32° 


to 212° 


32° 


to 392° 


32° 


to 572° 


32° 


to 212° 


32° 


to 572° 


32° 


to 212° 


32° 


to 572° 



0-00088420 
0-00275482 
0-00086133 
0-00184502 
0-00303252 
0-00118210 
0-00440528 
0-00171820 
0-00564972 



According to TrougMon. 



Platinum 

Steel 

Steel wire, drawn. 

Copper 

Silver 



0-00099180 
0-00118990 
0-00144010 
0-00191880 
0-00208260 



From 32° to 217° according to Roy. 



Glass (tube) 

Glass (solid rod) 

Glass cast (prism of ). . . 
Steel (rod of) 

Brass (Hamburg) 

Brass (English) rod. . . . 
Brass (English) angular. 



0-00077550 
0-00080833 
0-00111000 
0-00114400 
0-00185550 
0-00189296 
0-00189450 



Yulgar. 

l 

6 1 7 



4 ^ 6 

1 
4:iS 
_L_ 
403 

1 
39 9 



1 
349 



322 

_J . 

1 000 



363 

_ J 

116 1 



389 

_J 

846 

I 

2 27 
_JL_ 
5 82 



T7 7 



840 
__J 

694 

1 
52 J 

1 
480 



_1 

1289 

L 

12 3T 
1 



62 8 

1 ^ 



DILATATION PRODUCED BY HEAT. 143 

Measure of the Force of Dilatation. — The force with which 
solid hodies dilate and contract is equal to that which would 
compress them through the space they have dilated, or to that 
which would stretch them through a space equal to the amount 
of their contraction. Now, as it has been shown to be a phys- 
ical law that in every substance whatever, the same expenditure 
of heat, with the same extremes of temperature, will generate 
the same amount of mechanical power, it will follow that the 
less a body expands with any given increase of temperature, the 
more forcible will be the expansion, since the force, multiplied 
by the space passed through, must, in every case be a constant 
quantity. 

Dilatation of Liquids. — The rate of expansion of liquids 
becomes greater as the temperature becomes higher, so that a 
mercurial thermometer, to be accurately graduated, should have 
the graduations at the top of the scale somewhat larger than at 
the bottom. It so happens, however, that there is a similar 
irregularity in the expansion of the glass bulb, but in an opposite 
direction ; and one error very nearly corrects the other. Ther- 
mometers are accordingly graduated by immersing the bulb in 
melting ice, and marking the point at which the mercury stands. 
The point at which the mercury stands when the bulb is im- 
mersed in boiling water is next marked, and the space between 
the two marks is divided into 180 equal parts, and the graduation 
is extended above the boiling-point and below the freezing, by 
continuing the same lengths of division on the scale. The 
increment of volume which water receives on being raised from 
32° to 212° is a^rd of its bulk at 32°. Mercury at 32° expands 
J th of its bulk at 32° by being raised to 212° ; and alcohol, by 
the same increase of temperature, increases in volume -gth of its 
bulk at 32°. 

Compression and Dilatation of Gases. — "When a gas or 
vapour is compressed into half its original bulk, its pressure is 
doubled ; when compressed into a third of its original bulk, its 
pressure is trebled ; when compressed into a fourth of its original 
bulk, its pressure is quadrupled ; and generally the pressure varies 
inversely as the bulk into which the gas is compressed. So, in 



144 



THEORY OF THE STE AM-EXGIXE . 



like manner, if the volume be doubled, the pressure is made one- 
half of what it was before — the pressure being in every case 
reckoned from 0, or from a perfect vacuum. Thus, if we take 
the average pressure of the atmosphere at 14 - 7 lbs. on the 
square inch, a cubic foot of air, if suffered to expand into twice 
its bulk hy being placed in a vacuum measuring two cubic feet, 
will have a pressure of 7'35 lbs, above a perfect vacuum, and 
also of 7'35 lbs. below the atmospheric pressure ; whereas, if the 
cubic foot be compressed into a space of half a cubic foot, the 
pressure will become 29'4 lbs. above a perfect vacuum, and 14 - 7 
lbs. above the atmospheric pressure. This law, which was first 
investigated by Mariorte, is called MorioUe's law. It has already 
been stated that a cubic foot of air at 82° becomes 1*365 cubic 
feet at 212 : , the pressure remaining constant; or if the volume 
be kept constant, then the pressure of one atmosphere at 32 3 be- 
comes 1*365 atmospheres, or a little over 1| atmospheres at 212". 
These two laws, which are of the utmost importance in all phys- 
ical researches, it is necessary fully to understand and remember. 
The rates of dilatation and compression for each gas are not pre- 
cisely the same ; but the departure from the law is so small as 
to be practically inappreciable. According to iL Eegnault, the 
dilatation under the same pressure, and the increase of pressure 
with the same volume of different gases when heated from 32° 
to 212 r , is as follows: — 

CO-EFEICIEXT3 OF DILATATION OP DIEPEEEXT OASES. 



Hydrogen 

Atmospheric air 

Xitrogen 

Carbonic oxide 

Carbonic acid 

iProtoxide of nitrogen. 

Sulphurous acid , 

Cvanoiren 



Pressure 


Dilatation 


under constant 


under constant 


Toami. 


:. res-sure. 


0-3667 


0-3661 


0-3665 


0-36-70 


0-3663 


u 


0-3667 


0-3669 


0-3688 


710 


0-3676 


0.3719 


0-3 S45 


0-3903 


0-38-29 


0.3S77 



TSe rates of dilatation vary somewhat with the pressure and 
temperature, and in the case of gases, which are more easily 



DILATATION PRODUCED BY HEAT. 145 

condensable into liquids, the rate of dilatation increases rapidly 
with the density ; whereas the effect of heat is to remove these 
irregularities, and to maintain more completely the condition of 
a perfect gas. 

If we take the dilatation of atmospheric air when heated 180°, 
or from 82° to 212°, at 0-367 as determined by K Kegnault, 
then the amount of expansion which it will undergo from each 
Increase of one degree in temperature will be 180th of 0*367 = 
180th of -j 3 ^ = t^oVoo- = T9"o« I 11 other words, air will bo 
enlarged ^-oth part of its bulk at 32° by being raised one degree 
in temperature. 

If the same quantity of air or gas be simultaneously submitted 
to changes of temperature and pressure, the relations between 
its volumes, pressures, and temperatures, will be expressed by 
the general formula — 

v 490 ± t p\ 
~v' ^ 490 ±1? X p" ' 

where t and t' express the number of degrees above or below 
32° at which the temperature stands, + being used when above 
and — when below 32°, and the pressures being expressed in the 
usual manner by p and p'. By this formula, the volume of a gas 
at any proposed temperature and pressure may be found, if its 
volume at any other temperature and pressure be given, or the 
same thing may be done by the following rule : — 

THE BULK OF A GAS AT 32° BEING KNOWN, TO DETEEMINE ITS 
BULK AT ANY OTHEE TEMPEEATUEE, THE PEESSUEE BEING 



CONSTANT. 



Rule. — Divide the difference between the number of degrees in 
the temperature and 32° by 490. Add the quotient to 1 if 
the te'mp>erature be above 82°, and subtract it from 1 if it be 
below 32°. Multiply the volume of the gas at 32° by the 
resulting number, and the product will be the volume of the 
gas at the proposed temperature. 

Example.!. — What volume will 1000 cubic inches of air at 
82" acquire by being heated to 1000° Fahrenheit? 
7 



146 



THE OUT OF THE STEAM-ENGINE. 



EXPANSION OF DRY AIR BY HEAT. 
[la the columns V. cf the following table are expressed in cubic inches the volumes which a thou 
Band cubic inches of air at 328 will have at the temperatures expressed in the columns T., the air being 
luppcsed to be maintained under the same pressure ] 



|_T. 


V. 


T. 


V. 


T. 


Y. 


T. 


V. 


T. 


V. 


—50 


882-7 


8 


951-0 


66 


1069.4 


124 


1187-8 


182 


1306-1 


—49 


834-7 


9 


953-1 


67 


1071-4 


125 


11S9-8 


183 


130S-2 


—48 


886-7 


10 


955-1 


68 


1073-5 


126 


1191-8 


184 


1310-2 


—47 


88S-8 


11 


9571 


69 


1075-5 


127 


1193-9 


135 


1312-2 


-4G 


840-8 


12 


959-2 


70 


1077-6 


128 


1195-9 


186 


1314-3 


—45 


842-8 


13 


961-2 


71 


1079-6 


129 


1198-0 


1S7 


1316-3 


—44 


844-9 


14 


963-3 


72 


1081-6 


130 


1200-0 


188 


131S-4 


—43 


846-9 


15 


965-3 


73 


1083-7 


131 


1202-0 


189 


1320-4 


-42 


849-0 


16 


967-3 


74 


1085-7 


132 


1204-1 


190 


13224 


—41 


851-0 


17 


969-4 


75 


10S7-8 


133 


1206-1 


191 


1324-5 


— 4C 


8531 


18 


971-4 


76 


1089-8 


134 


120S-2 


192 


13265 


—39 


855-1 


19 


973-5 


77 


1091-8 


135 


1210-2 


193 


1828-6 


—38 


8571 


20 


975-5 


78 


1093-9 


136 


12122 


194 


1330-5 


— 37 


859-2 


21 


9T7-6 


79 


1095-9 


137 


1214-3 


195 


1332-6 


—86 


861-2 


22 


979-6 


SO 


1098-0 


133 


1216-8 


196 


1334-7 


— 35 


863-3 


23 


981-6 


81 


1100-0 


139 


12184 


197 


1336-7 


—34 


865-3 


24 


988-7 


82 


1102-0 


140 


1220-4 


19S 


1338-8 


—33 


S67-3 


25 


985-7 


83 


1104-1 


141 


1222*4 


199 


1340-8 


32 


869-4 


26 


987-8 


84 


1106-1 


142 


1224-5 


200 


1342-9 


—31 


871-4 


27 


9S9-8 


85 


1108-2 


143 


1226-5 


201 


1344-9 


—30 


873-5 


23 


991-8 


86 


1110-2 


144 


1228-6 


202 


1346-0 


—29 


875-5 


29 


993-9 


87 


1112-2 


145 


1230-6 


203 


1349-0 


— 23 


877-6 


30 


995-9 


88 


1114-3 


146 


1282-7 


204 


1351-1 


—27 


879-6 


31 


99S-0 


89 


1116-3 


147 


1234-7 


205 


1353-1 


—20 


8S1-6 


32 


1000-0 


90 


1118-4 


148 


1236-7 


206 


1355-1 


—25 


8S3-7 


83 


1002-0 


91 


1120-4 


149 


12SS-8 


207 


1357-3 


—24 


8S5-7 


34 


1004-1 


92 


1122-4 


150 


1240-S 


208 


1359-3 


—23 


8S7-8 


35 


1006-1 


93 


1124-5 


151 


1242-9 


209 


1361-3 


o-2 


889-8 


36 


10GS-2 


94 


1126-5 


152 


1244-9 


210 


13634 


—21 


891-8 


37 


1010-2 


95 


1128-6 


153 


1246-9 


211 


1365-5 


—20 


893-9 


38 


1012-2 


96 


1130-6 


154 


1249-0 


212 


1367-6 


—19 


895-9 


39 


1014-3 


97 


1132-7 


155 


1251-0 


213 


1369-2 


—18 


898-0 


40 


1016-3 


93 


1134-7 


156 


1253-0 


214 


1871-4 


—17 


900-0 


41 


1018-4 


99 


1136-7 


157 


12551 


215 


1373-2 


—16 


902-0 


42 


1020-4 


100 


1138-8 


158 


1257-1 


216 


1375-5 


—15 


904-1 


43 


1022-4 


101 


1140-8 


159 


1259-2 


217 


1377-5 


—14 


906-1 


44 


1024-5 


102 


1142-0 


160 


1261-2 


218 


1379-6 


—13 


908-2 


45 


1026-5 


103 


1144-9 


161 


1263-3 


219 


1381-6 


—12 


910-2 


46 


102S-6 


104 


1147-0 


162 


1265-3 


220 


1883-7 


—11 


912-2 


47 


1030-6 


105 


1149-0 


162 


1267-3 


230 


1404-1 


—10 


914-3 


48 


1032-7 


106 


1151-0 


164 


1269-4 


240 


1424-5 


— 9 


916-3 


49 


1034-7 


107 


1153-1 


165 


1271-4 


250 


1444-9 


— S 


918-4 


50 


1036-7 


108 


11551 


166 


1273-5 


260 


1465-3 


— 7 


920-4 


51 


1038-8 


109 


1157-1 


167 


1275-5 


270 


1485-7 


— 6 


922-5 


52 


1040-8 


110 


1159-2 


168 


1277-5 


280 


1506-1 


— 5 


924-5 


53 


1042-9 


111 


1161-2 


169 


1279-6 


290 


1526-5 


— 4 


926-5 


54 


1044-9 


112 


1163-3 


170 


1281-6 


300 


1546-9 


p 

u 


928-6 


55 


1046-9 


113 


1165-3 


171 


1283-7 


400 


1751-0 


2 


980-6 


56 


1049-0 


114 


1167-8 


172 


1285-7 


500 


1955-1 


— 1 


932-7 


57 


1051-0 


115 


1169-4 


173 


12S7-S 


600 


2159-2 





934-7 


53 


1053-1 


116 


1171-4 


174 


1289-8 


700 


2368-3 


1 


986-7 


59 


10551 


117 


1173-5 


175 


1291-8 


800 


2567-3 


2 


933-8 1 


60 


1057-1 


118 


1175-5 


176 


1293-9 


900 


2771-4 


8 


940-8 


61 


1059-2 


119 


1177-6 


177 


1295-9 I 


1000 


2975-5 


4 


942-9 


62 


1061-2 


120 


1179-6 


173 


1298-0 1 


1500 


3997-9 


5 


944-9 


63 


1063-3 


121 


1181-3 


179 


1300-0 


2000 


5016-3 


6 


947-0 


64 


1065-3 


122 


1188-7 


180 


13020 1 


2500 


6036-7 


7 


949-0 


65 


1067-3 


123 


1185-7 


181 


1304-1 i 


8000 


7057-1 



DILATATION PRODUCED BY HEAT. 147 

The difference between 1000° and 32° is 968, which divided by 
490 — 1-9755, and this added to 1 = 2-9755. Then 1000 x 2-9755 
*= 2975*5, which will be the volume in cubic inches at 1000°. • 

Example 2.— What will be the volume of the above air at 2000° £ 

Here 2000 — 32 = 1968 which -f- by 490 — 4-0163, and this 
added to 1 = 5-0163. Finally, 5-0163 x 1000 = 5016-3, which 
will be the volume of the air in cubic inches at 2000°. 

The volume which 1000 cubic inches of air at 32° acquires at 
all the various temperatures between — 50° and 3000° is shown 
in the preceding table : 

Anothee Eule. — To each of the temperatures "before and after 
expansion add the constant number 459 : divide the greater 
sum by the lesser, and multiply the quotient by the volume 
at the loicer temperature, and the product will give the ex- 
panded volume. 

Example 1. — What will be the volume of 1000 cubic inches 
of air at 32° when heated to 212°, the pressure being without 
alteration ? 

212 + 459 

Here 32 + 4 59" = 1 ' 366 ' wnicn multiplied by 1000=1366, 

which will be the volume in cubic inches at 212°. 

Example 2. — If the volume of steam at 212° be 1696 times 
the volume of the water which produced it, what will the vol- 
ume be if the steam be heated to 250*3 degrees Fahrenheit, the 
pressure remaining constant ? 

Here by the rule 212+459=671 and 250-3+459=709-3°. 
Moreover, 709-3 divided by 671 and multiplied by 1696=1792*8, 
which will be the bulk which the 1696 measures of steam will 
acquire when heated to 250*3° out of contact with water, the 
pressure remaining the same as at first. 

If we take the co-efficient of expansion of a perfect gas be- 
tween 32° and 212° at 0*365 instead of 0*367, the expansion per 
degree Fahrenheit will be „fey of the total bulk=0*002027G 
per degree Fahrenheit, instead of ^-g-th, as supposed by the rule 
from which the table is computed. This is equivalent to start- 
ing from the point of absolute zero, or 461*2° below the zero of 
Fahrenheit; as 461 *2° + 32° =498*2' 



148 



THEORY OF THE STEAM-ENGINE, 



TABLE SHOWING THE MELTING POINTS OF VARIOUS BODIES, IN DE« 
GREES OF FAHRENHEIT'S THERMOMETER. 



Name of Substance. 



Platinum 

English wrought-iron . . . 
French " " ... 
Steel 

" another sample 
Cast-iron 

" manganese 

" brown, fusible 

" " very fusible. 

" white, fusible 

" " very fusible. . 

Gold (very pure) 

Gold coin 

Copper 

Brass 

Silver (very pure) 

Bronze 

Antimony 



Zinc 



Lead. 



Bismuth , 



Tin. 



Alloy, 5 parts tin 

1 part lead 
Alloy, 4 parts tin 

1 part lead 
Alloy, 3 parts tin 

1 part lead 
Alloy, 2 parts tin 

1 part lead 
Alloy, 1 part tin 

3 parts lead 



552 



Degrees Fahren. 


Experimentalist. 


3082° 


Clarke. 


2912 


Vauquelin. 


2732 


Pouillet. 


2552 


u 


2372 


u 

j 


2192 


(t 


2282 


« 


2192 


M 


2012 


a 


2012 


u 


1922 


a 


2282 


a 


2156 


a 


1922 


(C 


1859 


Daniell. 


1832 


Pouillet. 


1652 


(t 


810 


(< 


700 


Murray. 


705 


G. Morveau. 


680 


Pouillet. 


629 


Person. 


608 


Pouillet. 


590 


Irvine. 


518 


Person. 


509 


Ermann. 


505 


Pouillet. 


480 


Crichton. 


512 


G. Morveau. 


455 


Person. 


446 


Pouillet. 


442 


Crichton. 


433 


Ermann. 


381 


Pouillet. 


372 


(C 


387 


u 


460 


u 



MELTING POINTS OF SOLIDS. 



140 



TABLE SHOWING THE MELTING- POINTS OF Y^ PIOUS BODIES, IN DE- 
GREES of Fahrenheit's thermometer — continued. 



Name of Substance. 



Alloy, 3 parts tin 

1 part bismuth 
Alloy, 2 parts tin 

1 part bismuth 
Alloy, 1 part tin j 

1 part bismuth ) 
Alloy, 4 parts tin 

1 part lead 

5 parts bismuth 

Sulphur ■] 

Iodine 

Alloy, 2 parts lead ) 

3 parts tin > 

5 parts bismuth ) 
Alloy, 5 parts lead ) 

3 parts tin > 

8 parts bismuth ) 

Alloy, 1 part lead ) 

1 part tin >- , 

4 parts bismuth ) 

Soda , 

Potash j 

Phosphorus 

Stearic acid 

Wax bleached 

Wax unbleached 

Stearine 

Spermaceti 

Acetic acid 

Tallow 

Ice 

Oil of turpentine . -. 

Mercury 



Degrees Fahren. 


Experimentalist. 


392 


Pouillct. 


333-9 


u 


286-2 


<( 



246 

239 

237 

225 

212 



212 



201 



Person. 
Dumas. 
Pouiliet. 



194 


Gay-Lussa 


162 


a 


136 


Pouiliet. 


111-6 


PersoD. 


109 


Pouiliet. 


100 


Murray. 


158 


Pouiliet. 


154 


(< 


142 


c( 


143 


Person. 


120 


Pouiliet. 


109 


<( 


120 


M 


113 


(( 


92 


(I 


32 


t( 


14 


I' 


-38.2 


u 



150 THEORY OF THE STEAlI-ENGCvE. 

LIQUEFACTION. 

Solidity is an accident of temperature, as there is every rea- 
son to believe that there is no substance in nature which may 
not be melted, and even vaporised, by the application of power- 
ful heat. 

There are two incidents attending liquefaction that are wor- 
thy of special attention : the first is that the liquefaction al- 
ways tahes place at the same temperature in the case of the 
same substance, so that the melting-point may in fact be used 
as an index of temperature ; and the second is that during lique- 
faction the temperature remains fixed, the accession of heat 
which has been received during the process of liquefaction being 
consumed or absorbed in accomplishing the liquefaction, or in 
other words it has become latent. This heat is given out again 
in the process of solidification. TVater deprived of air and cov- 
ered with a thin film of oil may be cooled to 20° or 22° below 
the freezing-point. But on solidification the temperature will 
rise to the freezing-point. Each different substance has, under 
ordinary circumstances, its own particular melting-point ; but 
it is found that the electrical condition of a body affects its melt- 
ing-point, and that electricity will fuse bodies at a low tempera- 
ture which commonly require for their fusion a very high degree 
of heat. Thus, platinum may be melted or vaporised by an 
electrical current, even although the heat generated is small ; 
and a process for separating metals from their ores by the aid of 
electricity has been projected by using low temperatures, aided 
by electricity, instead of high degrees of heat. In Part XV. of 
Taylor's Scientific demons, page 432, there is a paper ' On the 
Incandescence and Fusion of Metallic Wires by Electricity,' by 
Peter Eiess, being the substance of a paper read before the 
Roj'al Society of Berlin ; and in this paper it is shown that 
electrical fusion and vaporisation may take place at tempera- 
tures far below those at which metals are red hot. This prop- 
erty of electricity promises to be of service in the arts both in 
rendering refractory bodies fusible and in enabling bodies to be 
melted at low temperatures, which might be injured in their 
qualities by a subjection to high degrees of heat. Thus wrought- 



LATENT IIEAT OF LIQUEFACTION. 



151 



a-on if heated to a very high temperature, is liable to be burnt, 
unless carefully preserved from contact with the air ; whereas 
by sending a current of electricity through it, fusion may be 
accomplished at a comparatively low temperature, and any in- 
jury to the metal may thus be prevented. The melting-points 
of some of the most important substances are given in the pre- 
ceding table. 

Latent Heat of Liquefaction. — Ice in melting absorbs as 
tnuch heat as would raise the temperature of the same weight 
of water 142*65°, or as would raise 142*65 times that weight of 
water 1 degree ; yet, notwithstanding this accession of heat, the 
ice, during liquefaction, does not rise above 32°. If the heat 
employed to melt ice was applied to heat the same weight of 
ice-cold water, it would heat it to the temperature of 142*65° + 
32°=174*65°. The following table shows the amount of heat 
which becomes latent in the liquefaction of various bodies — the 
unit of latent heat being the amount of heat necessary to raise 
the same weight of water 1 degree : 



TAELE SHOWING- THE HEAT WHICH BECOMES LATENT IN THE 
LIQUEFACTION OF VARIOUS SOLID BODIES, AS ASCERTAINED BY 
M. PEESON. 



Names of Substances. 



Points of 

Fusion 
Fahrenheit. 



Chloride of lime. . 
Phosphate of soda . 

Phosphorus 

Pees'-wax (yellow), 
P/Arcet's alloy. . . . 

Sulphur 

Tin 

Bismuth 

Nitrate of soda. . . . 

Lead 

Nitrate of potash. . 
Zin? 



83*3 
97"5 
111-6 
143*6 
204-8 
239-0 
455*0 
518*0 
590-9 
629*6 
642.2 
793.4 



Latent Heat 

for Unity of 

Weight. 



72*42 

120-24 

8-48 

78-32 

10-73 

16-51 

25*74 

22*32 

113-36 

9*27 

83-12 

49-43 



By this table we see that the heat which becomes latent in 
melting a pound of bees' wax would raise the temperature of a 



152 THEORY OF THE STEAM-ENGINE. 

pound of water 7S*32 degrees; and the heat which becomes 
latent in melting a pound of lead would raise the temperature 
of a pound of water 9 '2 7 degrees. 

"When there is no external source of heat, from which the 
heat which becomes latent in liquefaction can be derived, and 
the circumstances are, nevertheless, such as to cause liquefaction 
to take place, the heat which becomes latent is derived from the 
substances themselves, and correspondingly lowers their temper- 
atures. Thus, when snow and salt are mixed together, the 
snow and salt are dissolved. But, as in melting they absorb 
heat, and as there is no external source from which the heat is 
derived, the temperature of the mixture falls very much below 
that of either of the substances before mixing. So, also, when 
saltpetre and other salts are dissolved in water, cold is produced, 
and on this principle the freezing mixtures are compounded 
which are employed to produce artificial cold in warm climates. 
A more effectual process, however, is to compress air, which 
heats it ; and the superfluous heat being got rid of by water, it 
will follow that when this air is again expanded, it will take 
back an amount of heat equal to that which it before lost, and 
which demand for heat may be made to cool surrounding bodies. 
A very effectual freezing machine is constructed on this princi- 
ple. But it is material that the air in expanding should be made 
to generate power, else the friction consequent on its escape will 
generate heat. 

VAPORISATION". 

As the first phenomenon of the application of heat to a solid 
eubstance is to dilate it, and the next to melt it, so also the fur- 
ther application of heat converts it from a liquid into a vapour 
or gas. The point at which successive increments of heat, in- 
stead of raising the temperature, are absorbed in the generation 
of vapour, is called the loiling-point of the liquid. Different 
liquids have different boiling-points under the same pressure, 
and the same liquid will boil at a lower temperature in a va- 
cuum, or under a low pressure, than it will do under a high 
pressure. As the pressure of the atmosphere varies at different 



LATENT HEAT OF VAPORISATION. 



153 



altitudes, liquids will boil at different temperatures at different 
altitudes, and the height of a mountain may be approximately de- 
termined by the temperature at which water boils at its summit. 

Difference between Gases and Vapours. — Vapours are sat- 
urated gases, or gases are vapours surcharged by heat. Ordi- 
nary steam is the saturated vapour of water, and if any of the 
heat be withdrawn from it, a portion of the water is necessarily 
precipitated. This is not so in the case of a gas under ordinary 
conditions. But if the gas be forced into a very small bulk, so 
that much of the heat is squeezed out of it, then it will follow 
that any diminution of the temperature will cause a portion of 
the gas to condense into a liquid. Surcharged or superheated 
steam resembles gas in its qualities, and a portion of the heat 
may be withdrawn from such steam, without producing the pre- 
cipitation of any part of its constituent water. 

Liquefaction of the gases. — Many of the gases have already 
been brought into the liquid state, by the conjoint agency of cold 
and compression, and all of them are probably susceptible of a 
similar reduction by the use of means sufficiently powerful for 
the required end. They must, consequently, be regarded as the 
superheated steams, or vapours, of the liquids into which they 
are compressed. The pressures exerted by some of these steams 
or gases are given in the following table : — 



TABLE SHOWING THE TEMPEEATUEE AND PEESSUEE AT WHICH 
THE SEVEEAL GASES NAMED AEE LIQUEFIED. 



Names of Gases condensed. 



Sulphurous acid . . . 

Cyanogen gas 

Hydi iodic acid.. . . 
Amuioniacal gas. . . 
Hydrochloric acid.. 
Protoxide of azote 
Carbonic acid ..... 



Temperature 
in degrees 
Fahrenheit. 


Pre69nre in 
Atmospheres. 


Temperature 
in degrees 
Fahrenheit. 


Pressure in 
Atmospheres. 


32° 


1-5 


46-4° 


2-5 


32 


2-3 






32 


4-0 






32 


4'4 


50 


5 


32 


8-0 






32 


37-0 


51-8 


43 


32 


32-0 


50 


45 



Latent lieat of Evaporation. — It has already been stated, that 
when a liquid begins to boil, the subsequent accessions of heat 

7* 



151 THEOKT OF THE STEA3I-ENGINE. 

which it receives go not to increase the temperature, but to ac- 
complish the vaporisation. The heat which thus ceases to be 
discoverable by the thermometer is called the Latent heat of 
Vaporisation ; and experiments have shown, that if the heat 
thus consumed had been employed to raise the temperature of 
the water, instead of boiling it away, the temperature of tha 
water would have been raised about 1,000 degrees Fahrenheit, 
or it would have raised about 1,000 times the same weight of 
water that is boiled off 1 degree Fahrenheit. 

The heat consumed in evaporating the same weight of dif- 
ferent liquids varies very much, but it does not follow that any 
of them would, therefore, be better than water as an agent for 
the generation of power, as the bulk of the resulting vapour in 
those which require least heat is small, in the p x>portion of the 
smaller quantity of heat expended in accomplishing the evapora- 
tion. Under the pressure of one atmosphere, or M - 7 lbs. per 
square inch, the latent heat of steam from water has been found 
to be 966\L Alcohol, which boils at 172*2, has a latent heat of 
evaporation of 364'3. Ether, which boils at 95°, has a latent 
heat of evaporation of 162 '8°, and sulphuret of carbon, which 
boils at 114 # 8°, has a latent heat of evaporation of 156°. 

The most important of the researches in connection with this 
subject are those which have reference to the Latent heat of 
Steam, and this topic has been illustrated by the researches of 
various experimentalists. At the atmospheric pressure, and 
starting at the temperature of 212°, the following estimates of the 
latent heat of steam have been formed bv the best authorities : — 



Watt 950- 

Southern 945* 

Lavoisier 1000' 

Eumford 1003- 



Despretz 955'S° 

Regnault 966-1 

Fabr8and I 961-8 

Siibermann ) 



The experiments which are generally considered to be the 
most correct in connection with this subject are those of M. 
Regnault. The following table, taken from his results, show 
that there is a difference of about 150° between the total heat 
of the vapour of water at the pressures corresponding to 32° 
and 44G° respectively. 



LATENT HEAT OF STEAM. 



155 



SENSIBLE AND LATENT HEAT OR STEAM. BY M. REGNAULT. 



i Temperature 

in degrees 
1 Fahrenheit. 

! 


Latent Heat. 


Sum of Sensible 

and 

Latent Heats. 


Temperature 
in degrees 
Fahrenheit. 


Latent Heat. 


Sum of Sensible 

and 
Latent Heats. 


32 


1092-6 


1124-6 


248 


936-6 


1187-6 


50 


1080-0 


1130-0 


266 


927-0 


1193-0 


68 


1067-4 


1135-4 


284 


914.4 


1198-4 


86 


1054-8 


1140-8 


302 


901-8 


1203-S 


104 


1042-2 


1146-2 


320 


889-2 


1209-2 


122 


1029-6 


1151-6 


338 


874-8 


1212-8 


140 


1017-0 


1157-0 


356 


862-2 


1218-2 


158 


1004-4 


1162-4 


374 


849-6 


1223-6 


176 


991-8 


1167-8 


392 


835-2 


1227-2 


194 


979-2 


1173-2 


410 


822-6 


1232-6 


212 


966-6 


1178-6 


428 


808-2 


1236-2 


230 


952-2 


1182-2 


446 


795-6 


1241-6 



Rules for connecting the temperature and elastic force of 
saturated steam. — Various formulas have been at different times 
propounded for deducing the elastic force of saturated steam 
from its temperature, and the temperature from the elastic force. 
The experiments of Mr. Southern, which were made at the in- 
stance of Boulton and "Watt, led to the adoption of the follow- 
ing rules, which, though not quite so accurate as some others 
which have since been arrived at, are sufficiently so for practical 
purposes, and being intimately identified with engineering prac- 
tice, it appears desirable to retain them. 



THE TEMPERATURE OE SATURATED STEAM BEING- GIVEN IN DEGEEES 
FAHRENHEIT, TO FIND THE CORRESPONDING ELASTIC FORCE IN 
INCHES OF MERCURY BY SOUTHERN'S RULE. 

Rule. — To tlie given temperature add 51*3 degrees. From the 
logarithm of the sum subtract the logarithm of 135*767, 
which is 2-1327940. Multiply the remainder by 5*13, and to 
the natural number answering to the sum, add the constant 
fraction '1. The sum will be the elastic force in inches of 
mercury. 

Example. — If the temperature of saturated steam be 250*3° 
Fahrenheit, what will be the corresponding elastic force in 
inches of mercury? 



15G THEORY OF THE STEAM-ENGINE . 

Here 250-3 x 51-3 = 301-6 Log. 2.4794313 

135*767 Lo?. 2-1327940 subtract 



remainder 0-3456373 
multiply by 5 -13 



Natural number 60*013 Lo<r. 1-7782493 



This natural number increased by "1 gives us 60*113 iuchea 
z: mercury, as the measure of the elastic force sought. 



THE ELASTIC FOECS OE 8ATEEATED STEAM BEIXQ G-IVEX IN" DTdlES 
OF MEECEEY, TO FIND THE COEEE3POXDIXG- TE.MBEEATUEE IN 
DEGEEES FAHEEXHEIT BY SOCTHEEX's EELE. 

Rele. — From the given elastic force subtract the constant frac- 
tion "1 ; divide the logarithm of the remainder by 5*13, and 
to the quotient add the logarithm 2'1327940. Find the 
natural number answering to the sum of the logarithms, and 
from the number thus found subtract the constant 51*3. The 
remainder icill be the temperature sought in degrees Fah- 
renheit. 

Example. — If the elastic force of saturated steam balances a 
vertical column of mercury 238*4 inches high, -what is the tem- 
perature of that steam? 

Here 23S-4 — 0*1 = 238-3 

Log. 238*3 =- 2*3771240 ^5 13 = 0*4633770 

2*1327940 add 



Natural number 394*61 Log. 2*5961710 

Constant 51*3 subtract 



Required temperature 343*31 degrees Fahrenheit. 



The temperature of the steam "which "will balance such a 
column of mercury, has been ascertained by observation to be 
343*6 decrees. 



TEMPERATURE AND PRESSURE OF STEAM. 



157 



Experiment* have been made by the French Academy, the 
Franklin Institute in America, and various other experimental- 
ists, to determine the elastic force of steam at different tempera- 
tures ; but of all these experiments, the most elaborate and the 
most widely accepted are those of M. Regnault. The results 
obtained by the French Academy are given in the following 
table, and those obtained by the Franklin Institute are very 
similar: — 



PEESSEEE OF STEAM AT DIEFEEENT TEMPEEATUEES. 

Remits of Experiments made by the French Academy. 

An atmosphere is reckonod as being equal to 29*922 inches of mercury. 



Pressure in 
Atmospheres. 


-Temperature in 
degrees of 
Fahrenheit. 


Pressure in 
Atmospheres. 


Temperature in 
degrees of 
Fahrenheit. 


1 


212° 


13 


386-66° 


H 


234 


14 


386-94 


2 


250-5 


15 


392-86 


24 


263-8 


16 


398-48 


1 3 


275-2 


17 


403-83 


H 


285 


18 


408-92 


4 


293-7 


19 


413-78 


*i 


300-3 


20 


418-46 


5 


307-5 


21 


422-96 


g_i 


314-24 


22 


427-98 


6" 


320-36 


23 


431-42 


H 


326-26 


24 


435-56 


7 


331-7 


25 


439-34 


n 


336-86 


30 


457-16 


8 


341-78 


35 


472-73 


9 


350*78 


40 


486-59 


10 


358-88 


45 


499-14 


11 


366-85 


50 


510-6 


12 


374 


i 





Formulee for connecting the temperature and elastic force of 
steam have been given by Young, Tredgold, Prony, Biot, Roche, 
Magnus, Holtzmann, Eankine, Eegnault, and many others — all 
more or less complicated. Regnault employs different formula) 



158 THEORY OF THE STEAM-EXGINE. 

for different parts of the therniometric scale, as appears from 
the following recapitulation in -which all the degrees are degrees 
centigrade : — 

eegxattlt's Fobmula foe the tempeeatfee and elastic 
foece of steam. 

Between 0° and 100° the fornmla is j 

Log. Y=a+d aP x — c/3^, 
which resembles the formula previously given by M. Biot. In 
this formula t is counted from 0° centigrade. «=4'7384380 ; Log. 

a x =0-006865036; Log. 1= 1- 996 7249 ; Log. 5=2-1340339, and 
Log. c=0'6116485. 

Between 100° and 230°, the formula he used is 
Log. F=<z — 5 a 7 — e-j3 r , 
in which t=^+20, t being the centigrade temperature reckoned 
from0°. Hence «=6-2640348; Log. a =1-994049292; Log. 
8= 1-998343862 ; Log. 5=0-1397743, and Log. c=0*6924351. 

The principal properties of saturated steam as deduced from 
the experiments of M. Regnault, exhibiting the pressure, the 
relative volume, the temperature, the total heat, and the weight 
of a cubic foot of steam of different densities, are given by Mr. 
Clark in the following tables : — 



REGXAULT S EXPERIMENTS OX STEAM. 



159 



PROPERTIES OF SATUEATED STEAM. 



BY 31. EEGXAITLT. 



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o 
Lbs, 


tt 






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Z5s. 




Fahr. 


Fahr. 


Lbs. 




Fahr. 


Fahr. 


Lbs. 


15 


1669 


213-1 


1178-9 


•0373 


48 


573 


278-4 


1198-8 


•1087 


16 


1572 


216-3 


1179-9 


•0397 


49 


562 


279-7 


1199-2 


•1108 


17 


1487 


219-5 


1180-9 


•0419 


50 


552 


281-0 


1199-6 


•1129 


18 


1410 


222-5 


1181-8 


•0442 


51 


542 


2S2-3 


1200-0 


•1150 


19 


1342 


225-4 


1182-7 


•0465 


52 


532 


283-5 


1200-4 


•1171 


20 


1280 


228-0 


1183-5 


•0487 


53 


523 


284-7 


1200-8 


•1192 


21 


1224 


230-6 


1184-3 


•0510 


54 


514 


285-9 


1201-1 


•1212 


22 


1172 


233-1 


1185-0 


•0532 


55 


506 


287-1 


1201-5 


•1232 


23 


1125 


235-5 


1185-7 


•0554 


56 


498 


288-2 


1201-8 


•1252 


24 


1082 


237-9 


1186-5 


•0576 


57 


490 


289-3 


1202-2 


•1272 


25 


1042 


240-2 


1187-2 


•0598 


58 


482 


290-4 


1202-5 


•1292 


26 


1005 


242-3 


1187-9 


•0620 


59 


474 


391-6 


1202-9 


•1314 


27 


971 


244-4 


1188-5 


•0642 


60 


467 


292-7 


1203-2 


•1335 


28 


939 


246-4 


1189-1 


•0684 


61 


460 


293-8 


1203-6 


•1356 


1 29 


909 


248-4 


1189-7 


•0686 


62 


453 


294-8 


1203-9 


•1376 


30 


831 


250-4 


1190-3 


•0707 


63 


447 


295-9 


1204.2 


•1396 


31 


855 


252-2 


1190-8 


•0729 


64 


440 


296-9 


1204-5 


•1416 


32 


830 


254-1 


1191-4 


•0751 


65 


434 


298-0 


1204-8 


•1436 


33 


807 


255-9 


1192-0 


•0772 


66 


428 


299-0 


1205-1 


•1456 


34 


785 


257-6 


1192-5 


•0794 


67 


422 


300-0 


1205-4 


•1477 


35 


765 


259-3 


1193-0 


•0815 


68 


417 


300-9 


1205-7 


•1497 


36 


745 


260-9 


1193-5 


•0837 


69 


411 


301-9 


1206-0 


•1516 


37 


727 


262-6 


1194-0 


•0858 


70 


406 


302-9 


1206-3 


'1535 


, 38 


709 


264-2 


1194-5 


•0879 


71 


401 


303-9 


1206-6 


"1555 


39 


693 


265-8 


1195-0 


•0900 


72 


396 


304-8 


1206-9 


•1574 


40 


677 


267-3 


1195-4 


•0921 


73 


391 


305-7 


1207-2 


•1595 


41 


661 


268-7 


1195-9 


•0942 


74 


386 


306-6 


1207-5 


•1616 


42 


647 


270-2 


1196-3 


•0963 


75 


381 


307-5 


1207-8 


•1636 


43 


634 


271-6 


1196-8 


•09S3 


76 


377 


308-4 


1208-0 


•1656 


44 


621 


273-0 


1197-2 


•1004 


77 


372 


309-3 


1208-3 


•1875 


1 45 


608 


274-4 


1197-6 


•1025 


78 


368 


310-2 


1208-6 


•1696 


46 


595 


275-8 


1198-0 


•1046 


79 


364 


311-1 


1208-9 


•171G 


1 47 


584 


277-1 


1198-4 


•1067 


80 


359 


312-0 


1209-1 


•1736 



iGO 



THEOET OF THE STEAM-ENGINE. 



PEOPEETLE3 OF SATEEATED STEAM —continued. 



BY M. EEG^ATTLT. 



! ■- 
j = 'l 

CD M 

22 

fl i 

o 


o 
"o 
> 
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Temperature. 


sS 

>— i 
>— i 


Weight of ono Cubic 
Foot. 


S 
p. 

o — 

u « 

GG — H 

a; 9 

-Z 2 

_, o< 
~o 

Lbs. 


© 

a 

p 

"el 
•o 

M 


CS 

o 

a 

o 
E-t 


o 
>— 1 
1— 1 

•« 

o 


o 

© 

a . 

'3 


Lbs. 




Fahr. 


Fahr. 


Lbs. 




Fahr.\ Fahr. 


Lbs. 


81 


355 


312-8 


1209-4 


•1756 


114 


261 


337-4 1 1216-8 


•2388 


82 


351 


313.6 


1209-7 


•1776 


115 


259 


338-01217-0 


•2406 


83 


343 


314-5 


1209-9 


•1795 


116 


257 


338-61217-2 


•2426 


84 


344 


315-3 


1210-1 


•1814 


117 


255 


339-3 11217-4 


•2446 


85 


340 


316-1 


1210-4 


•1833 


118 


253 


339-91217-6 


•2465 


86 


337 


316-9 


1210-7 


•1852 


119 


251 


340-5 1217-8 


•2434 


87 


ooo 
CJ-J-J 


317-8 


1210-9 


•1871 


120 


249 


341-lil2180 


•2503 


88 


330 


318-6 


1211-1 


•1891 


121 


247 


341-8 1218-2 


•2524 


89 


326 


319-4 


1211-4 


•1910 


122 


245 


342-4 1218-4 


•2545 


90 


323 


320-2 


1211-6 


•1929 


123 


243 


343-0 1218-6 


•2566 


91 


320 


321-0 


1211-8 


•1950 


124 


241 


343'6;i218-7 


•2587 


92 


317 


321.7 


1212-0 


•1970 


125 


239 


344-2'l2189 


•2608 


93 


313 


322-5 


1212-3 


•1990 


126 


238 


344-8 1219-1 


•2626 


94 


310 


323-3 


1212-5 


•2010 


127 


236 


345-4 ! 1219-3 


•2644 


95 


307 


324-1 


1212-8 


•2030 


128 


234 


346-01219-4 


•2662 


90 


305 


324-8 


1213-0 


•2050 


129 


232 


346-6 1219-6 


•2680 


97 


302 


325-6 


1213-3 


•2070 


130 


231 


347-2 1219-8 


•2698 


98 


299 


326-3 


1213-5 


•2089 


132 


228 


348-3 1220-2 


•2735 


99 


296 


327'1 


1213-7 


•2108 


134 


225 


349-5 1220-6 


•2771 


100 


293 


327-8 


1213-9 


•2127 


136 


222 


350-61220-9 


•2807 


101 


290 


328-5 


1214-2 


•2149 


138 


219 


351-8:1221-2 


•2846 


102 


288 


329-1 


1214-4 


•2167 


140 


216 


352-9 1221-5 


•2885 


103 


285 


329-9 


1214-6 


•2184 


142 


213 


354-0I1221-9 


•2922 


104 


283 


330-6 


1214-8 


•2201 


144 


210 


355-0 1222-2 


•2959 


105 


281 


331-8 


1215-0 


•2218 


146 


208 


356-1 


1222-5 


•2996 


108 


278 


331-9 


1215-2 


•2230 


148 


205 


357-2 


1222-9 


•3033 


107 


276 


332-6 


1215-4 


•2258 


150 


203 


358-3 


1223-2 


•3070 


108 


273 


333-3 


1215-6 


•2278 


160 


191 


363-41224-8 


■3263 


109 


271 


334-0 


1215-8 


•2298 


170 


181 


368-21225-1 


•3443 


110 


269 


334-6 


1216-0 


•2317 


180 


172 


372-91227-7 


•3623 


111 


267 


3353 


1216-2 


•2334 


190 


164 


377-51229-1 


•3S00 


112 


265 


336-0 


1216-4 


•2351 


200 


157 


381-7 1230-3 


•3970 


1113 

1 


263 


336-7 


1216-6 


•2370 






1 





regnault's experiments on vapours. 



1G1 



M. Regnaulfc extended his researches to the pressure of other 
vapours, beside that of water. The following are the results 
he obtained with alcohol, ether, sulphuret of carbon, chloroform, 
and essence of turpentine : 



TEMFERATURE AND ELASTIC FORCE OF THE VAPOUKS OF DIFFER- 
ENT LIQUIDS. BY M. EEGNAULT. 

[A millimetre is one thousandth part of a metre, or 0-03937 of an inch J 



1 

Teasioo of the Va- 
pour of Alcohol. 


Tension of the Va- 
pour of yEther. 


Tension cf the Va- 
pour of Sulphuret 
of Carbon. 


Tension of Vapour 

of Chloroform 

by Tension in 

Vacuo. 


i ] 

Tension of the Va- 
pour of Essence 
of Turpentine. 


t- 4 -a 


S £ 


&4 


M6 


to 


s^ 


R.-S 


S£ 


R.-S 


S£ 




~ s 


a 2 


~ 5 


Or= 


si 


0% 


^ u 


<0 


5 3 


— ' to 
g'C 


U § 


"2 <*> 


23 © 


<D ^ 


§s 


a> g 


5 2 

1*5 CO 


a 
CD <u 


gg 


-S 3 


s§ 


— O 


fi» 




*% 


3-a 
-5 -a 


aS 




.2^ 


£o 


ii"o 


S* 


I- o 


gfa 


«<£ 


^ 


S'S 


gs 


£"3 


S<§ 


£ g 


p*s 


3 a, 


P.' 




Oh£ 


3 w 


p<S 


3 m 


32 




S 2 

5 60 




gS 


Si 


§ 3j 




SE 


m "** 


Eh M 


ft 


H 


ft 


H 


ft 


H 


PM 


H 


Ph 


—21° 


3-12 


—20 


69-2 


—1 


5° 5S-8 


+ 10° 


130-4 


0° 


2-1 


—20 


8-34 


—10 


113-2 


— 1( 


) 79-0 


20 


190-2 


10 


2-3 


—10 


6-50 





1S2-3 


( 


) 127-3 


SO 


276.1 


20 


4-3 





12-73 


10 


2S6-5 


1( 


) 199-3 


36 


342.2 


80 


7-0 


10 


24-08 


20 


434-8 


2( 


) 29S-2 






40 


11-2 


20 


44 


30 


637-0 


3( 


) 434-6 


by the 


method 


50 


17-2 


30 


78-4 


40 


913-6 


4( 


) 617-5 


of ebu 


llition. 


60 


26-9 


40 


134-1 
220-3 


50 
60 


126S-0 
1730-3 


5( 
6' 


) 852-7 
) 1162-6 






70 

80 


419 
61-2 


50 


. • ■ • 





60 


350-0 


70 


2309-5 


7( 


) 1549-0 


36 


313-4 


90 


91-0 


70 


539-2 


80 


2947-2 


9( 


) 2030-5 


40 


364-0 


100 


131-9 


80 


812-8 


90 


3S99-0 


S( 


) 2623-1 


50 


524-3 


110 


187-8 


90 


1190-4 


100 


4920-4 


10( 


) 3321-3 


60 


73S-0 


120 


257-0 


100 


1685-0 


101 


7076-2 


IK 


) 4136-3 


70 


976-2 


130 


347-0 


110 


2351-8 




• • • 


• • > • 


12( 


) 5121-6 


80 


1367-8 


140- 


462-3 


120 


3207-8 






• . . . 


13< 


) 6260-6 


90 


1811-5 


150 


604-5 


130 


4331-2 






• • . • 


13< 


5 7029-2 


100 


2354-6 


160 


777-2 


140 


5637-7 




. . • 






.... 


110 


3020-4 


170 


989-0 


150 


7257-8 






• • ■ • 






120 


8818-0 


ISO 


1225-0 


152 


7617-3 




... 


.... 


... 





180 


47210 


190 
200 
210 
220 


1514-7 
1865-6 
2251-2 
2690-3 










... 













222 


2778-5 



Unit of heat. — It is convenient with the view of enabling ua 
to compare the quantities of heat in different bodies to fix upon 
some thermal unit, by which quantities of heat may be measur 
ed ; and the thermal unit employed in this country is the quan 



162 THEORY OF THE STEAM-ENGrNE. 

tity of heat which is required to raise a pound of pure water at 
its point to maximum density, through one degree Fahrenheit. 
In France the thermal unit employed is the quantity of heat re- 
quired to raise a kilogramme of pure water ct its point of great- 
est density through one degree Centigrade. A kilogramme is 
2-20-162 lbs. avoirdupois, or a pound avoirdupois is 0*453593 of 
a kilogramme. A degree Centigrade is 1*8 degrees Fahrenheit ; 
and a degree Fahrenheit is 0*555 of a degree Centigrade. There 
are 3*96832 British thermal units in a French thermal unit, 
and there is 0*251998 of a French thermal unit in a British 
thermal unit. 



SPECIFIC HEAT. 

The specific Ji-eat of a substance is an expression for the quan- 
tity of heat in any given weight of it at a certain temperature, 
just as its specific gravity is an expression for the quantity of 
matter in a given bulk. Specific heat is most conveniently ex- 
pressed by a reference to the number of thermal units consumed 
in producing a given elevation of temperature in the body under 
consideration ; or, if the weight of a heated body immersed in 
water be multiplied by the temperature it loses, and the weight 
of water be multiplied by the temperature it gains, the quotient 
obtained hj dividing the latter product by the former, will be the 
specific heat of the body. The specific heats of various substan- 
ces have been experimentally ascertained and recorded in tables, 
in which the specific heat of water is reckoned as unity. Thus, 
the specific heat of air is '2379, or it is 4*207 times less than that 
of water. An amount of heat, therefore, which would raise a 
pound of water 1 degree, would raise a pound of air 4*207 
degrees. 

The following tables of specific heats are derived from the 
experiments of the best authorities, and chiefly from those of 
M. Eegnault. The specific heat of ice is given on the authority 
of 3X. Person. 



SPECIFIC HEATS OF DIFFERENT SUBSTANCES 



163 



SPECIFIC! HEATS OF SOLIDS. 

The specific heat of water oeing reckoned as unity. 



NAME OF SUBSTANCE. 



Snecific 
Heat. 



Iron 

Cast-iron (white). 

Steel, soft , 

" tempered . , 

Copper , 

Brass , 

Zinc 

Lead 

Tin 

Silver 



0-11379 

0-12983 
0-11650 
0-11750 
0-09515 
0-09391 
0-09555 
0-03140 
0-05623 
0-05701 



NAME OF SUBSTANCE. 



Gold 

Platinum . . 

Glass 

Sulphur . . . 

Silicia 

Carbon. . . . 

Coke 

Diamond. . . 
Phosphorus 
Ice 



Specific 
Heat. 



0-03244 
0-03243 
0-19768 
0-20259 
0-19132 
0-24111 
0-20200 
0-14687 
0-18870 
0-50400 



SPECIFIC HEATS OF GASES AND VAPOTTBS. 

The specific heat ofioater oeing reckoned as unity. 



NAME OF GAS OK VAPOUR. 



Oxygen 

Nitrogen 

Hydrogen 

Chlorine 

Protoxide of nitrogen 

Binoxicle of nitrogen 

Carbonic oxide , 

Carbonic acid 

Sulphuret of carbon 

Sulphurous acid 

Ammonia , 

Protocarburet of hydrogen- 

(marsh gas) 

Bi-carburet of hydrogen 

Water vapour, or steam 

Alcohol vapour 

Oilier vapour 

Chloroform vapour 

Turpentine vapour 



Specific Heat. 


Densities. 






For equal 


For equal 




Weights. 


Volumes. 




0-2182 


0-2412 


1-1056 


0-2440 


0-2370 


0-9713 


3-4046 


0-2356 


0-0692 


0-1214 


0-2962 


2-4400 


0-2238 


0-3413 


1-5250 


0-2315 


0-2406 


1-0390 


0-2479 


0-2399 


0-9674 


0-2164 


0-3308 


1-5290 


0-1575 


0-4146 


2:6325 


0-1553 


0-3489 


2-2470 


0-5080 


0-2994 


0-5894 


0-5929 


0-3277 


0-5527 


0-3694 


0-3572 


0-9672 


0-4750 


0-2950 


0-6210 


0-4513 


0-7171 


1-5890 


0-4810 


1-2296 


2-5563 


0-1568 


0-8310 


5-3000 


0-5061 


2-3776 


4-6978 



164 



THE OUT OF THE STEAil-ENGIXE. 



SPECIFIC HEATS OF XIQUTDS. 

The specific heat of water deing reckoned as unity. 



NAME OF LIQT7TD. 


Specific i 
Heat. 


>- a:\ie of liquid. 


Specific 
Heat. 


Gin 

Olive Oil 


0-0333 
0-4672 1 
0-4770 j 
0-3096 ! 

i 


Solution Chlo. Lime . . 
Spirit of Wine at 97.. 


4684 
0-6448 
0-6588 
0-6501 





It will be observed from the foregoing tables that the specific 
heat of steam is nearly the same as the specific beat of ice. The 
specific beat of water, and also of air, occupying the same volume, 
Is found to be the same at all temperatures between boiling and 
freezing, and the specific beat of air under a constant pressure 
may be taken at 0-2379. In otber words, it requires just the 
same amount of beat to raise water and air one degree in tem- 
perature at any one part of tbe tbermometric scale as at any 
ether ; and tbe beat required to beat a pound of air 1 degree is 
only '2379, or less than one-fourth of the quantity required to 
beat a pound of water one degree. If therefore a pound of wa- 
ter at 60° bas transferred to it tbe beat in a pound of air at 
1000°, tbe water will not acquire as mucb elevation of tempera- 
ture as tbe air loses, but only -2379 of that temperature. 



EATIO OF SPECIFIC HEATS OF GASES TTXDEE COXSTAXT PEES3FEE 
TO THE SPECIFIC HEATS ENDEE CONSTANT YOLEaCE. 

"When air is compressed it generates beat, as is sbown \n 
tbe syringe in whicb a piece of tinder is ligbted by the beat pro- 
duced by the sudden compression of air; and, contrariwise, 
wben air or any otber gas is expanded it produces cold. When, 
therefore, a cubic foot of air of the atmospheric pressure is heat- 
ed until its pressure is doubled, it will have a certain tempera- 
ture which will fall if tbe air is suffered to expand into a volume 
of two cubic feet, and to restore tbe previous temperature more 
beat must be added. Tt will take more heat, therefore, to heat 



SPECIFIC HEATS AND SPECIFIC GRAVITIES. 



1G5 



A cubic foot of air to a given temperature, if it be suffered to ex- 
pand, tban if it be not suffered to expand ; and only that part 
of the heat is, properly speaking - , specific heat, which is shown 
by the rise of temperature, that which is absorbed in enlarging 
the volume being, in point of fact, latent heat. Both kinds of 
heat, however, are very generally called specific heat, but as the 
quantities are very different, it follows that there are two kinds 
of specific heat — the one the specific beat under a constant vol- 
ume, and the other the specific heat under the increased volume 
to which, the body naturally enlarges. It is only in the case of 
gases that there is a material difference between these specific 
heats. But in the case of gases the difference is very consider- 
able, and it is found that the specific heat under a constant press- 
ure divided by the specific heat under a constant volume, is 
equal, in the case of air, to 1*408 ; or, in other words, the speci- 
fic heat of air under a constant pressure is 1*408 times greater 
tban that of air under a constant volume. The specific heat of 
air under a constant pressure, may be taken at *2379, which 
makes the specific heat under a constant volume *169. The fol- 
lowing tables of the specific heats, and some other properties of 
solids, liquids, and gases are given by Air. Eankine : — 



SPECIFIC HEATS AXD SPECIFIC GEAVITIES OF METALS. 



Xame of lletal. 


Weight of a 

cubic loot in lbs. 

Do. 


Specific 

Gravity. 

S. G. 


Expansion 

Irom 

32? to 212*. 

E. 


Specific 

Keat 

C. 


Specific Heat 
in foot-pounds. 


Brass 


457 to 533 

524 

537 to 556 

1186 to 1224 

444 

480 

712 

1311 to 1373 

655 

- 490 

465 

436 

575 


7-S to 8-5 

8-4 

8-6 to 8-9 

19* to 19-6 

7-11 

7-69 

11-4 

21 to 22 

10-5 

7 - S5 

7-4 

7-2 


•00216 

•00181 

•001S4 

•0015 

•0011 

•0012 

•0029 

•U009 

•002 

•0012 

•0022 

-00294 


■0951 
•0298 

•1138 
■0293 
•0314 
•0557 

•0514 
•0927 


73-3 
23 

87-8 
22-6 
24-2 
43-0 

39-7 
71-6 


Bronze 


Copper 


Gold 


Iron, wrought . . 
Lead 


Platinum 

Silver 


fiteel 


Tin 


. Zinc 




|lce 


092 1 


•504 i 389 



iGo 



imoi.7 or :zz tIz^x-zv:-:: _ z. 



SPECIFIC HEAT? A2ZD SPECIFIC GP. A MILKS OF LIQUIDS. 



> zzs.i ::' 1:^1 



:. pure atSS 1 . . . 
44 sea, ordini.77 . . 

Z" ::'-;!. - ~_rr , 

" : : : : : _ : : 

.— .'. 7. . .". 

Ka -■— 

Nap] 7':, 

OiL Linseed 

ave , 

'• Wbal b 

* nf Trapes oe 

1 . .: :".r"nrr: 



68435 

:4- •: 

z ■: : 

57-18 

^■'. 
:-: ia 

aS I 

"" ■:: 

57"62 

.-4 : : 
a a 



i .. 

■:,: 

0-716 

:; •; : 
:•:■:■: 

..: 

: :" 



- - . . 



. ■:-_:;:•; 



:-: = 



::: . 









DSSSD3ES, \ OLDMESj EAXHB TZ SXPASSS03T, AM) SPBCIFIC 

zz_r~ :z sasbb. 









Zxi 11- 


_r_7rri JiZr. 




:-^a*cf Ga». 


fcot m 1 | |t 

i 7. 


:'- i. 
EI' : 

E. 


7: :.: 


Zi:- 

- Z:'. Z17 

". : - : = _:t 
Cp. 


Uadar I TToda 

: : " -i : :•-=:: - ; 

Kp. 


. 


'.' . 17- 

: 


365 

-36T 

■3©* 


■: 
:". 

. ;■:-:* 

0-173 


. ■-'.-. 
:■::- 

9369 

0-244 


:■: : 
i: . 

: r • 

: 

133-6 


153-45 

. -r'T 

i :■: 

371-3 
121-6 

1670 

:'■-.■. 
1:: i 


_!.:'_ ■■: -;~ :~r. . 

1 • . '. . . . .2 

Carbonic acid 

:!-: 

[ : 

Z-~ ~-i ... 

Yapoor of Mer- 


0-12259* 
012344 

■ " " 
. W84L1 


4-67S* 

: :"- 
;•: : 

12 36 

: . m 


i-:iteriii * 


_i L'-lrri 


■ -.--. :ri '.: 


: :— : _:ri 


: : - 1 i ie: 


I :■_ Li: . - 


::s -.:-:- 


:i:. 



La Oiese tables £he yolmnes are taken at Hie temperatore of 
■ l :\-.-- . ice, : : : 1 : : z : ept in the case of water, irhicli is taken 
at the lemperz: : t : maxumnn lensty, or B9 m l^. The pressure 
is taken at the zsnal atmospheric pressure of 2116*1 lbs. upon 
ft z}iiare foot. 

D i? the densitv c»r weight of 1 cubic foot of Uie substance iu 



SPECIFIC HEATS IN FOOT-FOUNDS. 1G7 

bs. avoirdupois under the pressure of one atmosphere, or 2116*4 

bs. on the square foot. 

V is the volume in cubic feet of 1 pound avoirdupois of the 
substance at the foregoing temperature and pressure. S.G-. is 
the specific gravity, water being taken as unity. E is the expan- 
sion of unity of volume for fluids, and unity of length for solids, 
at the temperature of melting ice, in being raised from the tem- 
perature of melting ice to the temperature of boiling water un- 
der the pressure of one atmosphere. is the specific heat in de- 
grees Fahrenheit, the specific heat of water being reckoned as 
unity, and C y is the specific heat under a constant volume, while 
O p is the specific heat under a constant pressure. K is the speci- 
fic heat, reckoned not in degrees of temperature, but in the equiv- 
alent value of pounds raised 1 foot high. It ha3 already been 
explained that there is as much power in the form of heat ex- 
pended in raising a pound of water 1 degree in temperature as 
would raise 772 lbs. to the height of 1 foot ; and 772 foot-pounds 
is, consequently, the mechanical equivalent of a pound of water 
raised 1 degree. Now as the specific heats of all bodies are de- 
terminable by the temperature to which a pound of the sub- 
stance will raise a pound of water, and as the accession of heat 
which a pound of water receives is transformable into its equiv- 
alent amount of mechanical power, it follows that the specific 
heats of all bodies may be represented by the amount of me- 
chanical power in foot-pounds, which is the equivalent of the 
heat consumed in raising a pound of any of these bodies through 
one degree of temperature. Such specific heats, accordingly, are 
those represented in the tables by the letter K ; the expression 
K v being the specific heat in foot-pounds of unity of weight under 
a constant volume, and K p the specific heat of the same weight 
under a constant pressure. The value of K p ~ K v , Mr. Kan kino 
states, is in the case of air, 1*408 ; oxygen, 1*4; hydrogen, 1*410 ; 
nitrogen, 1*409; and steam, considered as a perfect gas, 1*304; 
or, in other words, the specific heat under a constant volume is 
to the specific heat under a constant pressure as 1 to 1*4 in 
the case of oxygen, differing slightly \:\ the case of the other 
gases. 



168 THEORY OP THE STEAM-ENGINE. 

PHENOMENA OF EBULLITION. 

Influence of Viscosity or Molecular Attraction. — Salts dis- 
solved in water will raise the temperature of its boiling-point. 
The attraction of a salt for water being greater than the attrac- 
tion of the particles of the water for one another, will resist the 
repellent force of the heat to some extent. Mechanical pressure 
applied to the water has the same operation. Hence, water 
boils in a vacuum at a lower temperature than under the 
pressure of the atmosphere, and it also boils at a lower tem- 
perature under the pressure of one atmosphere than under a 
pressure of several atmospheres. "Water, which has been well 
purged of air by boiling, does not pass into the state of steam 
when heated in clean glass vessels, until it has attained a tem- 
perature considerably higher than its ordinary boiling point ; 
and when the steam finally forms, it forms rather by a jumping 
motion, or by a sudden shock, than by a gradual and silent dis- 
engagement. M. Magnus found that water well cleared of air 
may be raised to a temperature of 105° or 106° Centigrade 
before boiling, if the glass vessel in which it was heated were 
perfectly clean ; but if the vessel were soiled, or if dust or 
other foreign particles were suffered to enter it, the temperature 
would fall to the usual boiling point of 100° Centigrade. The 
sides of metallic vessels, or sawdust, metal filings, or insoluble 
particles of almost any kind, introduced into a liquid, lower its 
boiling-point. These particles are not at every point completely 
moistened by the water, and they have a less attraction for the 
particles of the fluid than the particles of the fluid have for one 
another. In the process of ebullition, therefore, the steam 
chiefly forms around those particles and seems to come out of 
them, and the boiling-point is lowered by the greater facility 
they occasion to the disengagement of the steam. M. Donny, 
by freeing water carefully from air, succeeded in raising it to a 
temperature of 135° without boiling ; but at this temperature 
steam was suddenly formed, and a portion of the water was 
projected forcibly from the tube. M. Donny concludes, from 
his experiments, that the mutual force of cohesion of the r&rti- 



SPHEROIDAL CONDITION OF LIQUIDS. 169 

cles of water is equal to a pressure of about three atmospheres, 
and to this strong cohesive force he attributes the irregular 
jumping motion observed in ebullition, and also some of those 
explosions of steam-boilers which heretofore have perplexed en- 
gineers. It is well known that cases have occurred in which an 
open pan of boiling water has exploded, producing fatal results, 
and such explosions cannot be explained on the usual hypothesis. 
M. Donny says that liquids by boiling lose the greater part of the 
air which they hold in solution, and therefore the molecular at- 
traction begins to manifest itself in a sensible manner. The 
liquid consequently attains a temperature considerably above 
its normal boiling-point, which determines the appearance of 
new air-bubbles, when the liquid separates abruptly, a quantity 
of vapour forms, and the equilibrium is for the moment restored. 
The phenomenon then recurs again with increased violence, and 
an explosion may eventually ensue. 

Steroidal Condition of Liquids on Hot Surfaces. — If a drop 
of water or other liquid be thrown upon a hot metal plate or 
other highly heated surface, it does not moisten the surface or 
diffuse itself over it, but forms a flattened ellipsoidal mass ; and 
if the drop be sufficiently small, it forms a minute spheroid, 
which revolves rapidly round a shifting axis, and evaporates 
very slowly without entering the state of ebullition. From 
Church's experiments it appears that it is necessary for the 
liquid to emit vapour before it can assume the spheroidal state. 
Molten lead dropped upon a very hot platinum plate did not as- 
sume the spheroidal state, whereas mercury dropped upon this 
plate assumed the spheroidal state at once. The most remark- 
able experiments, however, which have been made in illustration 
of the phenomena of the spheroidal state are those of M. Bou- 
tigny, and to him engineers are mainly indebted for calling their 
attention to the subject. One of the most singular results ob- 
tained by M. Boutigny is the power of making ice in a red hot 
crucible. A. small crucible or capsule of platinum being made 
white hot, some anhydrous sulphurous acid in the liquid state is 
poured into it. The boiling-point of this liquid is as low as 14° 
Fahrenheit ; but as it immediately on being projected into the 
8 



170 THEOEY OF THE iSTEAM-EXGINE. 

capsule assumes the spheroidal state, it remains upon the white 
hot metal without touching it ; and if a few drops of water he 
now let fall upon the liquid acid, the water will he immediately 
frozen, and a piece of ice may he turned out of the crucible. 
!M. Boutigny has also shown that if acids and alkalies in solutior 
he poured into a clean red hot platinum crucible they will no 
unite, but both will assume the spheroidal state and roll about 
the bottom of the crucible without entering into combination. 
Not merely the gravitation of the liquid, therefore, but also its 
chemical affinity, appears to be superseded by the causes which 
make it assume the spheroidal state. 

When a liquid assumes the spheroidal state it does not wet 
the surface, but appears to avoid touching it, like water sprinkled 
upon grease. Instead of entering into violent ebullition when 
it reaches the hot surface, its temperature will rise very little, 
and the drops of liquid will either remain at rest or will acquire 
a gyratory motion. When the surface is cooled down to 400° 
to 500°, depending on the nature of the surface and also on the 
nature of the liquid, the liquid will begin to diifuse itself, and 
will be suddenly scattered in all directions. The requisite tem- 
perature of a platinum plate to make water at the boiling-point 
assume the spheroidal state is 120° Centigrade, or 248° Fahren- 
heit ; but if glass be used instead of platinum, the temperature 
must be raised to 180° Centigrade, or 324° Fahrenheit. For 
water at 0° Centigrade, the temperatures required are 400° and 
800° respectively. 

"When water assumes the spheroidal state, it is possible by 
placing the eye on the level of the hot surface to see between 
the surface and the liquid. The electric circuit, moreover, is in- 
terrupted, showing that there is no actual contact between the 
liquid and the plate. The repulsion existing between the liquid 
and the plate is usually imputed to the existence of an atmos- 
phere of vapour upon which, as upon a cushion, the spheroids 
are supposed to rest. There is no reason to conclude, however, 
because vapour is raised from a liquid, that therefore its gravity 
must be suspended, and the cause is rather to be sought for in 
the motion of the spheroid, or of its internal particles, whereby 
the motion to which gravity is due is partially counteracted. 



COMMUNICATION OF HEAT. 171 

Spheroidal State of tlie Water in Boilers. — There can be. 
no doubt that the water of boilers is sometimes repelled from 
the metal in the same manner as would be done if it were in 
the spheroidal state, and explosions have, no doubt, frequently 
had their origin in this phenomenon. Land boilers, whether of 
the cylindrical or waggon form, frequently bend down in the 
bottom where the strongest heat of the furnace impinges, and 
lead rivets, inserted in them for purposes of safety, are some- 
times melted out. The water appears to be repelled from the 
iron in those parts of the boiler bottom where the heat is great- 
est, and the iron becomes red hot, and is bagged or bent out by 
the pressure of the steam. In some boilers the bottom can at 
any time be made red hot by very heavy firing, and in most fac- 
tory boilers the bottom will be more or less injured if the stoker 
urges the fire very much. If gauge cocks be inserted at differ- 
ent levels, in a small upright cylindrical boiler, so that one cock 
is near the top, another near the bottom, and the rest in inter- 
mediate positions, it will follow, that if sufficient water be intro- 
duced into the boiler to show at the lowest gauge cock, it will 
continue to show there so long as a moderate heat is maintained. 
So soon, however, as the fire is made to burn fiercely, so as to 
impart a strong heat to the bottom, the water will disappear 
from the bottom cock and show in the top cock, thus proving 
that the water has been repelled by the heat until it occupies 
the top part of the boiler instead of the bottom part. 

COMMUNICATION OF HEAT. 

Heat may be communicated from a hot body to a cold one in 
three ways — by Eadiation, by Conduction, and by Circulation. 

The rapidity with which heat radiates varies, other things 
being equal, as the square of the temperature of the hot body in 
excess of the temperature of the cold one ; so that a body if made 
twice as hot will lose a degree of temperature in one-fourth of 
the time ; if- made three times as hot, it will lose a degree of 
temperature in one-ninth of the time ; and so on, in all other 
proportions. This explains how it comes that a very small pro- 
portion of surface in a boiler of which the furnace is maintained 



172 



THEORY OF THE STEAM-ENGINE. 



at a high temperature is equivalent to a much larger proportion 
of surface when the temperature is somewhat lower. Badiant 
heat may be concentrated into a focus by a reflector, in the 
same manner as light, and, like light, it may likewise be made 
to undergo refraction and polarisation. 

The conduction of heat through different substances varies 
v ery nearly in the same proportion as their conducting powers 
for electricity. Taking the conducting power of silver as 100, 
the following are the conducting powers of metals according ti 
the best authorities : — 

CONDUCTING POWEKS OF METALS. 



Name of Body. 


Conductivity for Electricity. 


Conductivity i 
for Heat. 


Eiess. 


Becquerel. 


Lenz. 


Wiedemann 
and Franz. 




100-0 
G6-7 
59-0 
18-4 

io-o 

12-0 

' V-6 

10-5 
5-9 


100-0 
91-5 
64-9 

1*4-6 
12-35 

8-2Y 
'7-93 


100-0 
73-3 
58-5 
21-5 
22-6 
13-0 

10-3 
1-9 


ioo-o ! 


Copper 


fS'6 i 


Gold 


53-2 ; 


Brass 


23-6 

14-5 

11-9 

11-6 

8-5 

8-4 

63 

1-8 


Tin 


Iron 


Steel 


Lead 


Platinum 


Bismuth 





The conducting power of marble is about the same as the 
conducting power of bismuth ; and the conducting powers of 
porcelain and bricks are each about half that of marble. The 
conducting power of water is very low, and hence heat is trans- 
mitted downwards through water only very slowly. But up- 
wards it is transmitted rapidly by virtue of the circulation which 
then takes place. 

The efficiency of the heating surface of a boiler will depend 
very much upon the efficiency of the arrangements which are 
in force for maintaining or promoting a rapid circulation of the 
water. In like manner, the rapidity of the circulation which is 
maintained in the water used for refrigeration in surface con- 



CONDUCTING POWERS OF DIFFERENT SUBSTANCES. 173 

densers will mainly determine the weight of steam condensed 
in the hour by each square foot of refrigerating surface. Peclet 
found by a number of experiments that water, when used as the 
refrigerating fluid, was about ten times more effectual than air; 
and he further found that when water was used for refrigera- 
tion, each square foot of copper surface was able to condenso 
about 21-^- lbs. of steam in the hour. Mr. Joule, however, found 
that a square foot of copper surface might, by maintaining a 
rapid circulation of the cooling water, be made to condense 100 
lbs. of steam in the hour — the cooling water being contained in 
a pipe concentric with that containing the steam, and flowing 
in the opposite direction. "With this rapidity of refrigeration, 
the cooling surface of a condenser need only be about one six- 
teenth of the heating surface of the boiler which supplies the 
engine with steam. In ordinary land boilers 10 square feet of 
heating surface will boil off a cubic foot, or 62|- lbs. of water in 
the hour ; and one square foot of heating surface will therefore 
boil off one-tenth of this, or 6*25 lbs. of water in the hour. To 
boil off 100 lbs. in the hour would at this rate require 16 square 
feet of heating surface. But the 100 lbs. of steam thus boiled off 
will, according to Mr. Joule, be condensed by one square foot of 
cooling surface ; so that, if this authority be accepted, the surface 
of a well-constructed condenser need only be about one-sixteenth 
of the heating surface of theboiler, the steam of which it condenses. 
The importance of maintaining a rapid circulation in the 
water of boilers has not yet been sufficiently recognised. It is 
desirable that solid water and not steam should be in contact 
with the heating surface, else the metal plating will be liable to 
become overheated, and any given area of heating surface will 
be much less effective. The species of boiler invented by Mr. 
David Napier, called the haystack boiler, and in which the 
water is contained in vertical tubes, is about the best species of 
boiler for keeping up a rapid circulation of the water. But it 
necessary to apply large return pipes or a wide water space all 
round the exterior of the boiler, with a diaphragm to permit 
ascending and descending currents, in order that the water car- 
ried upward by the steam may be immediately returned. 



L7-4 the'I'ET :j r:zz 5TZA3i-LXCrrsx. 



; ::::t-::;::. 

: mbustion is energetic chemical combination between the 
" s _en of the air and the constituents of the combustible. The 
combustibles chiefly used to generate the heat consumed by 
steam-engines are coal, wood, and sometimes charcoal. 

Coal consists chiefly of carbon and hydrogen, but the pro- 
portions in which these elements enter into the composition of 
different coals is very various. Cannel coal consists of about 60 
per cent, of volatile matter, and 40 per cent, of coke and earthy 
matter, whereas splint coal consists of about 65 per cent, of 
coke, and 35 per cent, of volatile matter. Air consists of oxy- 
gen and nitrogen, mixed in the proportions of 8 lbs. of oxyg si 
to every 23 lbs. of nitrogen, or 1 lb. of oxygen to svery I \ lbs. 
of nitrogen. To accomplish the combustion of 6 lbs. of carbon, 
16 lbs. of oxygen are necessary, forming 22 lbs. of carbonic 
icid, which will have the same volume as the oxygen, and, 
iherefore, a greater density. To accomplish the combustion of 
1 lb. of hydrogen, 8 lbs. of oxygen are necessary. When, there- 
fore, we know the proportions of carbon and hydrogen existing 
in coal, it is easy to tell the quantity of oxygen, and, conse- 
quently, the quantity of air necessary for its combustion. As a 
general rule, it may be stated that, for every pound of coal 
burned in a furnace, about 12 lbs. of air will be necessary to 
furnish the oxygen required, even if every particle of it entered 
into combination. But from careful experiments it has been 
found, that in ordinary furnaces, where the draught is produced 
by a chimney, about as much more air will in practice be neces- 
sary, or about 24 lbs. per lb. of coal burned. In the case of 
furnaces, with a more rapid draught maintained either by a 
steam jet or a fan blast, a smaller excess of air will suffice, and 
in those cases about 18 lbs. of air will be reqirired from the 
combustion of 1 lb. of coal. K a cubic foot of air weigh 1*291 
oz^ then 12 lbs. or 192 oz. will measure about 150 cubic feet, as 
1*291 oz. bears the same proportion to 1 cubic foot, as 192 oz. 
bears to 150 cubic feet nearly. In ordinary furnaces, with a 
tliimney therefor, which require 24 lbs. of air per lb. of coal, 



TOTAL HEAT PRODUCED BY COMBUSTION. 



175 



the volume of air necessary for the combustion of 1 lb. of coal 
will be about 300 cubic feet, which is equal to the content of a 
room measuring about 6 feet 8£ inches every way. 

The specific gravity of oxygen is a little more than that of 
air, being by the latest experiments 1-106, while that of air is 1. 
Now, as 16 lbs. of oxygen unite with 6 lbs. of carbon to rbrm 
22 lbs. of carbonic acid, and, as the volume of the carbonic acid 
at the same temperature remains only the same as that of the 
original oxygen, it follows that the density or specific gravity of 
the carbonic acid must be greater than that of the oxygen, in 
the same proportion in which 22 is greater than 16. Multiply- 
ing therefore 1*106, which is the specific gravity of oxygen, by 
22, and dividing by 16, we get 1*521, which must be the specific 
gravity of carbonic acid, if the specific gravity of oxygen is 
1*106. Formerly, the specific gravity of oxygen was reckoned 
at 1*111, but there is reason to believe that 1*106 is the more 
accurate determination. 

Total Heat of Cornbustion. — The temperature to which a 
pound of fuel would raise a pound of water, or the total heat 
of combustion in thermal units, has been carefully investigated 
by ~KM.. Favre and Silbermann, whose determinations are reca- 
pitulated and condensed by M. Kankine as follows : — 

TOTAL HEAT OF COMBUSTION OF 1 lb. OF EACH OF THE 
COMBUSTIBLES ENUMEEATED. 



Combustible, 
1 lb. of each, being burned. 

Hydrogen gas 

Carbon, imperfectly burned, 
so as to make carbonic 
oxide 

Carbon, completely bnrned, ) 
so as to make carbonic >• 
acid ) 

Oleliant gas 

Various liquid hydrocarbons . . 

Carbonic oxide, as mucb as") 
is made by the imperfect I 
combustion of 1 lb. of car- f 
bon, viz. 2 J lbs J 



Lbs. of 

Air 
required. 



2! 

3f- 

1\ 



Lbs. of 
Air 

required. 



Total Heat 

in Tbormal 

Units. 



12 

15?- 



62,032 
4,400 

14,500 

21,344 
j from 21,000 
( to 19,00<J 

10,100 



Evaporative 

Power 
from 212". 



C4-2 
4-55 

15-0 

22*1 
j from 22 
( to 20 

10-45 



176 THEOXY OF THE S1EAM-ENGLN1S 

With regard to the quantities stated as being the total heal 
of combustion respectively of carbon completely burned, carbon 
imperfectly burned, and carbonic oxide, !XIr. Eahkine says that 
the following explanation has to be made : — 

The burning of carbon is always complete at first ; that is to 
say, one pound of carbon combines with 2§ lbs. of oxygen, and 
makes 3f lbs. of carbonic acid ; and although the carbon is solid 
immediately before the combustion, it passes during the com- 
bustion into the gaseous state, and the carbonic acid is gaseous. 
This terminates the process when the layer of carbon is not so 
thick, and the supply of air not so small, but that oxygen in 
sufficient quantity can get direct access to all the solid carbon. 
The quantity of heat produced is 11,500 thermal units per lb. 
of carbon, as already stated. 

But in other cases part of the solid carbon is not supplied 
directly with oxygen, but is first heated, and then dissolved into 
the gaseous state, by the hot carbonic acid gas from the other 
parts of the furnace. The 8f lbs. of carbonic acid gas from 1 
lb. of carbon, are capable of dissolving an additional lb. of car- 
bon, making 4f lbs. of carbonic oxide gas ; and the volume of 
this gas is double of that of the carbonic acid gas which pro- 
duces it. In this case, the heat produced, instead of being that 
due to the complete combustion of 

1 lb. of carbon or . . . . • . 14,500 

falls to the amount due to the imperfect combustion of 2 lbs. 

of carbon, or . . . . 2 x 4,400 x 8,800 

Showing a loss of heat to the amount of . . 5,700 

which disappears in volatilising the second pound of carbon. 
Should the process stop here, as it does in furnaces ill supplied 
with air, the waste of fuel is very great, as the carbonic oxide — 
which is a species of invisible smoke — has a large quantity of 
carbon in it which is dissipated in the atmosphere ^thout use- 
ful result. But when the 4-f lbs. of carbonic oxide gas, contain- 
ing 2 lbs. of carbon, is mixed with a sufficient supply of fresh 
air, it burns with a blue flame, combining with an additional 2| 
lbs. of oxygen, making Ti lbs. of carbonic acid gas, and giving 



ECONOMIC VALUES OF DIFFERENT COALS. 



177 



additional heat of double the amount due to the combustion of 
1£ lb. of carbonic oxide ; that is to say, 

10,100x2 = 20,200 
to which being added the heat produced by the imperfect 

combustion of 2 lbs. of carbon, or 8,800 

thore is obtained the heat due to the complete combustion of 

2 lbs. of carbon, or . . . . 2 x 14,500 — 29,000 

The evaporative powers of different kinds of coal in practice 
is given in the following table : — 



TABLE SHOWING- THE ECONOMIC VALUES OF DIFFERENT COALS. 
BY DH LA BECHE AND PLAYFAIK. 



Names of Coal employed in the 
Experiments. 



Graigola 

Anthracite (Jones & Co.) 
Oldcastle Fiery Vein. . . 

Ward's Fiery Vein 

Binea 

Llangennech 

Pentrepoth 

Pentrefellin 

Dutfryn 

Mynydd Newydd 

Three-quarter Bock Vein 
Cwra Frood Bock Vein 

Cwm Nanty-gros 

Besolven 

Pontypool 

Bedwas 

Ebbw Vale 

Porthmawr 

Coleshill 



' Dalkeith Jewel Seam. . . 
" Coronation | 

Seam j 

g j Wallsend Elgin 

f /2 Fordel Splint 

[ Grangemouth 



mi j Broomhill 

£ ( Lydney (Forest of Dean) 

Slievardagh (Irish An- l 
thracite) J* 

Wylam's Patent Fuel. . . 
Warlich's " 

Bell's " 

~8* 



Economical 
evaporating 
power, or num- 
ber of lba., of 
Water evapo- 
rated from 
212° by 1 lb. of 
Coal. 



Weight of 
1 cubic foot 
of the Coal 
as used for 
Fuel. 

lbs. 



9-35 
9-46 

8-94 
9-40 
9-94 

8-88 
8-72 
6-36 

10-14 
9-52 
8-84 
8-70 
8-42 
9-53 
7-47 
9-79 

10-21 
7-53 
8-00 

7-08 

7-71 

8-46 
7-56 
7-40 

7-30 

8-52 

9-S5 

8-92 

10-36 

8-53 



60-166 

58-25 

50-916 

57-433 

57-08 

56-98 

57-72 

66-166 

53-22 

56-33 

56-388 

55-277 

56-0 

58-66 

55-7 

50 5 

53-3 

53-0 

53-0 

49 8 

51-66 

54-6 
55-0 
54-25 

52-5 
54-444 

62-8 

65-08 
69-05 
65-3 



Space occu- 
pied by 1 ton 

of the Coal 
in cubic feet. 



37-23 

38-45 

43-99 

39 

39-24 

39-34 

38-80 

33-85 

42-09 

89-76 

89-72 

40-52 

40-00 

38-19 

40-216 

44-32 

42-26 

42-02 

42-26 

44-98 

4336 

41-02 
40-72 
40-13 

42-67 
41-14 

35-68 

34-41 
32-44 
34-80 



Rate of eva- 
poration, or 
number of 
lbs. of Water 
evaporated 
per hour. 

Mean. 



441-48 
409-37 
464-30 
529-90 
486-95 
373-22 
381-50 
247-24 
409-32 
470-69 
486-86 
379-80 
404-16 
890-25 
250-40 
476-96 
460-22 
347-44 
406-41 

855-18 

37008 

435-77 
464-98 
380-49 

397-78 
487-19 

473-18 

418-89 
457-84 
549-11 



178 THEORY OF THE STE AIM-ENGINE. 

Maximum Temperature of the Furnace.— When we know 
the total heat of a combustible in thermal units, the weight of 
the smoke and ashes or the products of combustion, as they are 
called, and their specific heat, it is easy to tell what is the high- 
est temperature that the furnace can attain, supposing that the 
air is not artificially heated. Thus the chief products of combus- 
tion of coal being carbonic acid, steam, nitrogen, and ashes, with 
a certain proportion of residual air, which passes unchanged 
through the fire ; then, if we reckon the specific heat of carbonic 
acid at 0*217, of steam at 0*475, of nitrogen at 0*245, of air at 
0*238, and of ashes at 0*200, and take into account the quantities 
of each which are present, the mean specific heat of the prod- 
ucts of combustion may be taken, without much error, as about 
equal to the specific heat of air. Now, as 12 lbs. of air are re- 
quired for the combustion of a pound of carbon, even if every 
particle of the oxygen be supposed to enter into combination, 
the weight of the products of combustion will on that supposi- 
tion be 12 lbs. -f 1 lb., or 13 lbs. If we take the total heat of 
combustion of carbon or charcoal at 14,500, and the mean speci- 
fic heat of the products of combustion at 0*238, then the specific 
heat multiplied by the weight will be 3*094 ; and 14,500 divided 
by 3*094 = 4689, which will be the temperature to which the 
furnace would be raised in degrees Fahrenheit, supposing every 
atom of oxygen that entered the furnace entered into com- 
bination. If, however, as will be the case in ordinary furnaces, 
twice that quantity of air necessarily enters, then the weight of 
the products of combustion of 1 lb. of coal will be 25 lb., which, 
multiplied by the specific heat = 5*95, and 14,500 divided by 
5*95 =2,437, which is the temperature in degrees Fahrenheit 
that, on this supposition, the furnace would have. If 18 lbs. of 
air be supplied per lb. of coal, as suffices in the case of furnaces 
with artificial draught, then the weight of the products of com- 
bustion will be 19 lbs., which, multiplied by the specific heat, 
gives 4*522, and 14,500 divided by 4*522, gives 3,207 as the tem- 
perature of the furnace in degrees Fahr. This in point of fact 
may be taken as a near approach to the temperature of hot fur- 
naces such as that of a locomotive boiler. 



RATE OF COMBUSTION. 



179 



Tlio increased volume which any given quantity of air at 32° 
will acquire, by raising its temperature through any given num- 
ber of degrees, can easily be determined by the rule already 
given for that purpose. Mr. Eankine has computed the volume 
in cubic feet, which 12 lbs. of air, 18 lbs., and 24 lbs., will respec- 
tively acquire, when heated to different temperatures, by com- 
bining with 1 lb. of carbon in a furnace ; the volume of 12 lbs. at 
32°, and at the atmospheric pressure, being taken at 150 cubic 
feet, of 18 lbs. at 225 cubic feet, and of 24 lbs. at 300 cubic feet. 
The results are as follows : 

TEMPEEATUEES OF COMBUSTION AND VOLUMES OF PEODUCTS. 





Supply of Air in pounds per lb. of fuel. 


12 lbs. I 18 lbs. 


24 lbs. 


Temperatures. 


1 




Volume of Air or Gases in cubic feet at each 




Temperature. 


32° 


150 


225 


300 


68° 


161 


241 


322 


104° 


172 


258 


344 


212" 


205 


307 


409 


» 392° 


259 


389 


519 


1 5*72° 


314 


471 


628 


f 752° 


369 


553 


738 


1112° 


479 


718 


957 


1472° 


588 


882 


1176 


1832° 


697 


1046 


1395 


2500° 


906 


1359 


1812 


3275° 


1136 


1704 




4640° 


1551 







Bate of Combustion. — The rate of combustion, or the quan- 
tity of fuel burned in the hour upon each square foot of fire- 
grate, varies very much in different classes of boilers. In Cor- 
nish boilers it is 3£ lbs. per square foot ; in the older class of 
land boilers, 10 lbs.; in more recent land boilers, 13 to 14 lbs.; 
in modem marine boilers, 16 to 24 lbs., and in locomotive boilers? 
SO to 120 lbs. on each square foot of fire-grate in the hour. 



180 THEORY OF THE STEAM-ENGINE. 



THERMODYNAMICS. 

It lias been already stated that heat and power are mutually 
convertible, and that the power in the shape of heat which ia 
necessary to raise a pound of water through one degree Fahren- 
heit, would, if utilised without waste in a thermo-dynamic en- 
gine, raise 772 lbs. through the height of 1 foot. A pound of 
water raised through a degree centigrade is equivalent to 1390 lbs. 
raised through the height of 1 foot. In every heat engine, the 
greater the extremes of temperature, or the hotter the boiler or 
source of heat and the colder the condenser or refrigerator, the 
larger will be the proportion of the heat utilised as power. 

In a perfect steam engine, if a be the temperature of the 
boiler, reckoning from the point of absolute zero, and 5 be the 
temperature of the condenser, reckoning also from the point of 
absolute zero, the fraction of the entire heat communicated to 
the boiler which will be converted into mechanical effect, will 

be -. Eow it is clear if a = Z>, or if the temperature of the 

boiler and condenser are the same, the value of becomes 

a 
equal to 0, or there is none of the heat utilised as power, whereas, 

on the other hand, if a be taken larger and larger, the value of 
the fraction becomes continually greater, indicating that by in- 
creasing the difference of the temperatures of the boiler and 
condenser, a great quantity of the heat -expended is converted 
into mechanical effect, and by taking a= oo, or infinity, the limit 
to which the fraction approaches is found to be unity, showing 
that in such a case, if it were possible of realisation, the whole 
of the heat would be converted into power. 

The formula given by Professor Thomson for determining 
the power generated by a perfect thermo-dynamic engine, is as 
follows: — 

If S be the temperature of the source of heat, and T be the 
temperature of the refrigerator, both expressed in centigrade 
degrees ; and if H denote the total heat in thermal units centi- 
grade, entering the engine in a given time; and J be Joule's 



POWER PRODUCIBLE IN A PERFECT ENGINE. 1S1 

equivalent of 1390 lbs. raised one foot high by a centigrade de- 
gree ; — then the power produced, or TV" the work performed, is 

■^ -^^ S — T 
TV=JH * 

S + 274 

This formula may be expressed in words, as follows : — 

TO FIND THE POWEE GENEEATED BY A PEEFECT ENGINE IMPELLED 
EY THE MOTIVE POWEE OF HEAT. 

Rule. — From the temperature of the source or toiler, subtract 
the temperature of the condenser ; divide the remainder by 
the sum of the temperature of the source and 274, and multi- 
ply the quotient by the total heat communicated to the en- 
gine per minute, expressed in the number of degrees through 
which it would raise one pound of water. Finally, multiply 
this product by 1390. The result is the number of pounds 
that the engine will raise a foot high in the minute. The 
temperatures are all taken in degrees centigrade. 

Example. — In a steam-engine working with a pressure of 14 
atmospheres, the temperature of the steam in the boiler will be 
215° centigrade, and the temperature of the condenser may be 
taken at 44'44° centigrade. If a grain of coal be burned per 
minute, the heat imparted every minute to a pound of water 
will be -905° centigrade. Now 215 — 44-44° = 170-56 and 
215 -f 274 = 489, and 170'56 divided by 489 = 0*35, which mul- 
tiplied by '905 and by 1390 = 440 lbs. raised 1 foot high every 
minute, which as a grain of coal is burned every minute, is very 
nearly the same result as that before indicated. 

Cheapest Source of Motive Power. — The cheapest source of a 
mechanical power that will be available in all situations, is, so 
far as we yet know, the combustion of coal. Electricity and 
galvanism have been proposed as motive powers, and may be 
ised as such, but they are much more expensive than coal. Mr. 
Joule has ascertained by his experiments that a grain of zinc, 
consumed in a galvanic battery, will generate sufficient power to 
raise a weight of 145-6 lbs. through the height of one foot; 



182 THEOliY OF THE STEAH-EXGESE. 

whereas a grain of coal, consumed by combustion, will generate 
sufficient power to raise 1261*45 lbs. to the height of 1 foot. 

Moreover, it appears certain that Mr. Joule's estimate of the 
heating power of coal is too small. A pound of coal will, under 
favourable circumstances, evaporate 12 lbs. of water, which is 
equivalent to a pound of water being heated 2 degrees Fahren- 
heit by a grain of coal, or it is equivalent to 154i lbs. raised 
through 1 foot. This is more than ten times the power gener- ; 
&ted by a pound of zinc. But as thermo-electric engines, it is 
estimated, expend their energy about four times more bene- 
ficially than heat engines, the dynamic efficacy of a pound of 
zinc may be taken as about 4-lOths of that of a pound of cod. A 
ton of zinc, however, costs fifty or sixty times as much as a ton 
of coals, while it is not half so effective. There does not ap- 
pear, therefore, to be the least chance of heat engines being 
superseded by electro-dynamic engines, of which zinc or some 
other metal supplies the motive force. 

EXPANSION OF STEAM. 

When air is compressed into a smaller volume, a certain 
amount of power i3 expended in accomplishing the compression, 
which power, as in the case of a bent spring, is given back again 
when the pressure is withdrawn. If, however, the air when 
compressed is suddenly dismissed into the atmosphere, the power 
expended in compression will be lost ; and there is a loss of 
power, therefore, in dispensing with that power, which is re- 
coverable by the expansion of the air to its original volume. 
Xow the steam of an engine is in the condition of air already 
compressed; and unless the steam be worked in the cylinder 
expansively — which is done by stopping the supply from the 
boiler before the stroke is closed — there will be a loss of a cer- 
tain proportion of the power which the steam would otherwise 
produce. If the flow of steam to an engine be stopped when the 
piston has performed one-half of the stroke, leaving the rest of 
the stroke to be completed by the expanding steam, then tho 
efficacy of the steam will be increased 1*7 times beyond what it 



MODE OF COMPUTING BENEFIT OF EXPANSION. 183 

would have been had the steam at half-stroke been dismissed 
without extracting more power from it ; if the steam be stopped 
at one-third of the stroke, the efficacy will be increased 2*1 
times; at one-fourth, 2*4 times; at one-iifth, 2*6 times; at one- 
sixth, 2*8 times; at one-seventh, 3 times; and at one-eighth, 3*2 
times. 

TO FIND THE INCEEASE OF EFFICIENCY AKISLNG- FEOM WOKKINO 
STEAM EXPANSIVELY. 

Rule. — Divide tlie total length of the stroke oy the distance 
{which call 1) through ichich the piston moves oefore the steam 
is cut off. The Neperian logarithm of the whole stroke ex- 
pressed in terms of the part of the stroke performed with the 
full pressure of steam, represents the increase of efficiency 
due to expansion. 

Example 1. — Suppose that the steam be cut off at^-ths of 
the stroke : what is the increase of efficiency due to expansion ? 

Here it is plain that y^ths of the whole stroke is the same as 
T ?- of the whole stroke. The hyperbolic logarithm of 7*5 is 
2*015, which increased by 1, the value of the portion performed 
with full pressure, gives 3*015 as the relative efficacy of the 
steam when expanded to this extent, instead of 1, which would 
have been the efficacy if there had been no expansion. 

If the steam be cut of at |, f , ?, f , |-, £, or -£th of the stroke, 
the respective ratios of expansion will be 8, 4, 2*66, 2, 1*6, 1*33, 
and 1*14, of which the respective hyperbolic logarithms are 
2-079, 1-386, 0-978, 0'693, 0*470, 0*285, and 0*131 ; and if the 
steam be cut off at T V, T 2 o, T 3 o, T \, T 5 n -, j%, tt>- il, or T 9 n ths of the 
stroke, the respective ratios of expansion will be 10, 5, 3*33, 2*5, 
1*66, 1*42, 1*25, and I'll, of which numbers the respective 
hyperbolic logarithms are 2'303, 1*609, 1-203, 0-916, 0*507, 0*351, 
0*223, and 0*104. With these data it will be easy to compute 
the mean pressure of steam of any given initial pressure when 
cut off at any eighth part or any tenth part of the stroke, as we 
have only to divide the initial pressure of the steam in lbs. per 
square inch by the ratio of expansion, and to multiply the quo- 



184 THEORY OF THE STEAM-ENGINE. 

tient by the hyperbolic logarithm, increased by 1, of the number 
representing the ratio, which gives the mean pressure through 
out the stroke in lbs. per square inch. Thus, if steam of 100 lbs. 
be cut off at half stroke, the ratio of expansion is 2, and 100 
divided by 2 and multiplied by 1"693 = 84*65, which is the mean 
pressure throughout the stroke in lbs. per square inch. The 
terminal pressure i3 found by dividing the initial pressure by the 
ratio of expansion; thus, the terminal pressure of steam of 
100 lbs. cut off at half stroke will be 100 divided by 2 = 50 lbs. 
per square inch. 

Example 2. — "What is the mean pressure throughout the 
stroke of steam of 200 lbs. pei square inch cut off at y^th of the 
stroke ? 

Here 200 divided by 10 =■ 20, which, multiplied by 3'303 (the 
hyperbolic logarithm of 10 increased by 1) gives 06*04, which is 
the mean pressure exerted on the piston throughout the stroks 
in lbs. per square inch. 

If the steam were cut off at |th of the stroke instead of tV" n > 
then we should have 200 divided by 8 = 25, which, multiplied 
by 3*079 (the hyperbolic logarithm of 8 increased by 1), gives 
76 "975 lbs., which would be the mean pressure on the piston 
throughout the stroke in such a case. 

If the initial pressure of the steam were 3 lbs. per square 
inch, and the expansion took place throughout |ths of the stroke, 
or the steam were cut off at -|th, then 3 -j- 8 = *375, which 
x by 3*079 = 1*154625 lbs. per square inch of mean pressure. 

There are various expedients for stopping off the supply of 
steam to the engine at any desired point of the stroke, which 
are described in my ' Catechism of the Steam Engine,' and 
which, consequently, it would be superfluous to recapitulate 
here. One mode is by the use of an expansion valve, and 
another mode is by extending the length of the face of the or- 
dinary slide valve by which the steam is let into and out of the 
cylinder, which extension of the face is called lap or cover 
For the purposes of this vork it will be sufficient to recapitu- 
late the mean pressure of the steam on the piston of an engine 
throughout the whole stroke, supposing the steam to be cut off 



PRESSURES AT DIFFERENT RATES OF EXPANSION. 185 

at different successive points of the stroke, counting first by 
eighths, and next by tenths, and to explain what amount of lap 
answers to a given expansion, and what expansion follows the 
use of a given proportion of lap. The mean pressure of the 
steam throughout the stroke, with different initial pressures of 
steam and different rates of expansion, or, in other words, the 
equivalent constant pressure that would be exerted throughout 
the stroke if such a pressure were substituted for the varying 
pressures to which the piston is in reality subjected, are exhib- 
ited in the following tables, in one of which the pressures aro 
those which would ensue if the expansion took place during so 
many eighths of the stroke, and in the other during so many 
tenths of the stroke : — 



MEAN PEESSUEE OF STEA.M AT DIFFEKENT DENSITIES AND KATES 

OF EXPANSION. 

The column headed 0, .contains the Initial Pressure in lbs., and tlie 
remaining columns contain the Mean Pressure in, lbs., with different 
amounts of Expansion. 





Proportion of the Stroke through which Expansion takes 


place. 







1 


3 


3 


4 


.5 


e 


I 

7 ; 


s 


S 


S 


S 


s 


s 


8 ' 
] 


3 


2-96 


2-89 


2-75 


2-53 


2-22 


1-789 


1-1.54 


4 


3-95 


3-S5 


3-67 


3-38 


2-96 


2-386 


1-539 


5 


4948 


4-S18 


4-593 


4-232 


3-708 


2-982 


1-921 


6 


5-937 


5-782 


5-512 


5-079 


4-450 


3-579 


2-309 


7 


6-927 


6-746 


6-431 


5-925 


5-241 


4-175 


2-694 


S 


7*917 


7-710 


7-350 


6-772 


5-934 


4-772 


3-079 





8-906 


8-673 


8-268 


7-618 


6-675 


5-368 


3-483 


10 


9-S96 


9-637 


9-187 


S-465 


7-417 


5-965 


3-848 


11 


10-885 


10-601 


10-106 


9-311 


8159 


6-561 


4-233 


12 


1PS75 


11-565 


10-925 


10-158 


8-901 


7-158 


4-618 


13 


12-865 


12-528 


11-943 


11-004 


9-642 


7-7.54 


5-003 


14 


13-854 


13492 


12-862 


11-851 


10-3S4 


8-531 


5-388 


15 


14-S44 


14-456 


13-7S1 


12-697 


11-126 


8-947 


5-773 


16 


15-834 


15-420 


14-700 


13-544 


11-868 


9-544 


6-158 


17 


16-823 


16-383 


15-613 


14-390 


12-609 


10-140 


6-542 


18 


17-813 


17-347 


16-537 


15-237 


13-351 


10-737 


6-927 


19 


18-702 


18-311 


17-448 


16-803 


14093 


11-333 


7-312 


20 


19-792 


19-275 


18-375 


17-970 


14-S35 


11-930 


7-697 


25 


24-740 


24-093 


22-968 


21-162 


18-543 


14-912 


9-621 


30 


29-6SS 


2S-912 


27-562 


25-395 


22-252 


17S95 


11-546 


35 


34-636 


33-731 


33156 


29-627 


25-961 


20-877 


13-470 


40 


39-585 


38-550 


36-750 


33-860 


29-670 


23-S60 


15-395 


45 


44-533 


43-368 


41-343 


38-092 


33-378 


26-S42 


17-319 


50 


49-481 


48-1S7 


45-937 


42-325 


37-067 


29-825 


19-243 



183 



THEOBY OF THE SEE AM-E>7GC>~E . 



MEAX PBES3HK1 OF STEAIT AT DTFFEEEST DENSITIES AND EATES 

OF EXPANSION. 

The column headed contains the Initial Pressure in lbs., and the 
remaining columns contain the Mean ^Pressure in lis., iciih diferen* 
amounts of Expansion* 





Prrjportiea 


::" :':; ;tr:ii -j^rzish. — ii:Ji E 


^i^:- 


tskefi place. 







i 


2 


- 


« 


s 


8 


T 






ii 


lu 


To 


io 


Id 


: - 


IO 


& 


io 


s 


. - : 


' 2^8»3 


. m 


: rac 


2-539 


- - J 


l ::: 


1-663 


0-99*3 


4 


s-974 


3-913 


i ~: 


3-614 


3-356 


3-065 


2-642 


1 Hi 


1-320 


5 




4 : - 


i-ras 


i 518 


4-232 


! 33S 


£3d 


2-609 


1-651 


6 


5-961 


' ' ' 


- ' 


5-421 


5-079 


4-593 


3-963 


3-130 


1-951 


7 


• 6-955 


;-_: 


v 


6-325 


5-925 


5-364 




" -. 


2-311 


: 


" £ 


: >:: 


7! : 


' ..: 


". 


6-131 


- m 


4-174 


6-641 


9 


: »#2 


s : : 


: ' : 


5 :;: 


- as 




5-945 


i : : 


2-971 


10 


9"936 


: 184 


-/. 


> :>-: 


8-465 


- m 


6-606 


5-213 


S-S02 


11 


10-9-29 


: " : : 


10-395 


S-939 


9-311 


Mm 


7-266 


5-739 


8-632 


12 


11-9-23 


11-740 




ii as 


10-153 


9-196 


' .- 


6-261 


3-962 


13 


1- ::: 


12-719 


12-255 


11-746 


10-994 


9-963 


8-587 


6-753 


4-292 


14 


13*10 


11 " 


13-230 


12-650 


11-551 


10-729 


i MS 


■ I r 


4-622 


15 


14 4 


14-676 


14-175 


13-554 


:. m 


11 if i 




' :.' 


4-953 


16 


IS ": - 




15-13) 


14-457 


13-544 


12-2*2 


1 ■ 


■ m 


5-253 


IT 


16-8W 


16-632 


16-065 


: r : :: 


14t)51 


ii .; 


11-230 


i :-" 


5-613 


18 


1T-SS4 


17-611 


17-010 


16-264 


15-237 


n -;: 


1TS9I 


- 


5-944 


V 


13 378 


IS :: 


IT '": 


17-163 


16-083 


14-561 


12-551 


9-914 


6-273 


jj 


: 372 


19-565 


13- 


IS '. 


16* 


15-32S 


IS -2 12 


10-43G 


6-60) 


- T 


M '-- 


:- - ' 


.: .:: 


22-590 


21-162 


19-100 


16-515 


13-040 


S-255 


: 


. - - 


." '■'-'. 


i: ; r 


:*: : 


85395 


22-99*2 


:: 31S 


15-654 


9-906 


: ' 


34-TT-3 


14-344 


33i>75 


31-626 


1 :.' 


.: $24 


23T21 


15-263 


11-557 


-: 


i 144 


- - 


;- ■ : 


36-144 


>•: -:: 


30-656 


26-2*24 


. 572 


1*806 


45 


44-912 


44 .': 


42 -.' 


40-662 


": - 


M 388 


29-727 


23-451 


14-559 


afi 


,': V:. 


4&32fl 


-/ .: 


GriSH 


42 :.: 


. ; : . ■ 


33*30 


26-090 


16-510 



Example. — If steam "be admitted to the cylinder at a pressure 
of 3 lbs. p ri' £ josfq inch, and be suffered to expand during |th of 
the stroke, the mean pressure dnring the whole stroke will be 
2'98 lbs. per square inch. In like manner, if steam at the press- 
ure of 3 lbs. per square inch were cut off after the piston had 
gone through the £th of the stroke, lea\ing the steam to expand 
through the remaining fths, the mean pressure during the whole 
stroke would be 1*154 lb?, per square inch. 



BELAIION5 BI 



tB LAP OF THE VALYE 
OF EXPANSION. 



LSD THE AMOUNT 



The ruled for determining the relations between the lap of 
the yalve^and the amount of the expansion are as follows:- 



EFFECTS OF LAP 01< THE VALVE. 187 

TO FIND HOW MUCH LAP MUST BE GIVEN ON" THE STEAM SIDE, 
IN OEDER TO CUT THE STEAM OFF AT ANY GIVEN PAET OF 
THE STEOKE. 

Rule. — From the length of the stroke of the 'piston subtract the 
length of that part of the stroke that is to be made before 
the steam is cut off. Divide the remainder by the length of 
the stroke of the piston, and extract the square root of the 
quotient. Multiply the square root thus found by half the 
length of the stroke of the valve, and from the product take 
half the lead, and the remainder will be the amount of lap 
required. 

TO FIND AT WHAT PAET OF THE STEOKE ANY GIVEN AMOUNT OF 
LAP ON THE STEAM SIDE WILL CUT OFF THE STEAM. 

Rule. — Add the lap on the steam side to the lead : divide the 
sum by half the length of stroke of the valve. In a table 
of natural sines find the arc whose sine is equal to the quo- 
tient thus obtained. To this arc add 90°, and from the sum 
of these two arcs subtract the arc whose cosine is equal to the 
lap on the steam side divided by half the stroke of the valve. 
Find the cosine of the remaining arc, add 1 to it, and mul- 
tiply the sum by half the stroke of the piston, and the prod- 
uct is the length of that part of the stroke that will be made 
by the piston before the steam is cut off. 

TO FIND HOW MUCH BEFORE THE END OF THE STEOKE THE EX- 
HAUSTION OF THE STEAM IN FEONT OF THE PISTON WILL BE 
CUT OFF. 

Rile. — To the lap on the steam side add the lead, and divide 
the sum by half the length of the stroke of the valve. Find 
the arc whose sine is equal to the quotient, and add 90° to it. 
Divide the lap on the exhausting side by half the stroke of 
the valve, and find the arc whose cosine is equal to the quo- 
tient. Subtract this arc from the one last obtained, and 
find the cosine of the remainder. Subtract this cosine from 



188 THEORY OF THE STEAM-ENGINE. 

2, and multiply the remainder by half the stroke of the pis- 
ton. The product is the distance of the piston from the end 
of the stroke when the exhaustion is cut off. 



ro find how far the piston is feom the end of its stroke. 

WHEN THE STEAM THAT IS PROPELLING IT BY EXPANSION IS AL- 
LOWED TO ESCAPE TO TOE CONDENSER. 

Rule. — To the lap on the steam side add the lead ; divide the 
sum by half the stroke of the valve, and find the arc whose 
sine is equal to the quotient. Find the arc whose cosine is 
equal to the lap on the exhausting side, divided by half the 
stroke of the valve. Add these two arcs together, and sub- 
tract 90°. Find the cosine of the residue, subtract it from 1, 
and multiply the remainder by half the stroke of the piston. 
The product is the distance of the piston from the end of its 
stroke, when the steam that is propelling it is allowed to es- 
cape to the condenser. 

Note. — In using these rules all the dimensions are to be taken 
in inches, and the answers will be found in inches also. 

It will readily be perceived from a consideration of theso 
rules that — supposing there is no lead — the point of the stroke 
at which the steam is cut off is determined by the proportion 
which the lap on the steam side bears to the stroke of the valve. 
"Whatever the absolute dimensions of the lap may be, therefore, 
it will follow that, in every case in which it bears the same ratio 
to the stroke of the valve, the steam will be cut off at the same 
point of the stroke. 

As some of the foregoing rules are difficult to be worked out 
by persons unacquainted with trigonometry, it will be conven- 
ient to collect the principal results into tables, which may bo 
applied without difficulty to the solution of any particular ex- 
ample. This accordingly has been done in the three following 
tables, the mode of using which it will now be proper to ex- 
plain. 



RELATIONS OF LAP AND EXPANSION. 



189 



-PROPORTION OF LAP REQUIRED TO ACCOMPLISH VARIOUS DEGREES OF 
EXPANSION. 



Distance of the piston " 
from the termina- 
tion of its stroke, 
when the steam is - 
cut off, in parts of 
the length of its 


21 

or 
1 


7 


e 

2 1 

01 

i 


s 

21 


4 
21 

or 

i 

G 


3 
21 

or 

i 

8 


2 

or 

i 

12 


i 
i 

21 


Lap on the steam side ~) 
of the valve, in de- (_ 
cimal parts of the f 
length of its stroke. J 


'239 


•270 


•250 


•228 


•204 


•ITT 


•144 


•102 



Example. — In the first line of the first table will be found 
eight different parts of the stroke of the piston designated ; and 
directly below each, in the second line, is given the quantity of 
lap requisite to cause the steam to be cut off" at that particular 
part of the stroke. The different amounts of the lap are given 
in the second line in decimal parts of the length of the stroke 
of the valve ; so that, to get the quantity of lap corresponding 
to any of the given degrees of expansion, it is only necessary to 
take the decimal in the second line, which stands under the frac- 
tion in the first, that marks that degree of expansion, and mul- 
tiply that decimal by the length we intend to make the stroke 
of the valve. Thus suppose we have an engine in which we 
wish to have the steam cut off when the piston is a quarter of 
the length of its stroke from the end of it, we look in the first 
line of the table, and we shall find in the third column from the 
left, £. Directly under that, in the second line, we have the 
decimal, '250. Suppose that we consider that IS inches will be 
a convenient length for the stroke of the valve, we multiply the 
decimal -250 by 18, which gives 4£. Hence we learn, that with 
an 18-inch stroke for the valve, 4£ inches of lap on the steam 
side will cause the steam to be cut off when the piston has still 
a quarter of its stroke to perform. 

Half the stroke of the valve should always be at least equal 
to the lap on the steam side added to the breadth * of the port ; 
consequently, as the lap in this case must be 4| inches, and as 

* By the 'breadth 1 of the port, is meant its dimensions in the direction of the 
valvc'3 motion : in short, its perpendicular depth when the cylinder is upright 



190 



IHEOBT OF THE SIXAH-EXGIH] 



half the stroke of the valve is 9 inches, the efficient brea l£h : 
the port cannot be more than 9 — 4^ = 44 inches, since half : if 
is covered over by the lap. If this breadth of port is not suffi- 
cient to give the required area to let the steam in and out, we 
must increase the stroke of the valve ; by which means we shall 
get both the lap and the breadth of the port proportionally in- 
ed. Thus, if we make the length of valve-stroke 20 inches, 
we shall have for the lap -250 x 20=5 inches, and for the breadth 
of the port 10 — 5 = 5 inches. 

This table, as we have already intimated, is computed on the 
supposition that the valve is to have no lead ; but, if it is to 
have lead, all that isnecessir - is to subtract half the proposed 
lead from the lap found from the table, and the remainder will 
be the proper quantity of lap to give to the valve. Suppose 
that, in the last example, the valve was to have -£ inch of lead, 
: should subtract ^ inch from the 5 inches, found for the lap 
by the table. This would leave 4J inches for the quantity : : 
lap that the valve ought to have. 

H. LAP IX IXCHE3 BEQUISED OS THE STEAM SIDE OP THE VAM.XE Z » COT 

THE STKAV OFF AT AXY OP THE UXDEE-SOTEP PARTS OF THE STROKE. 



LtL.:l 




Proportion of tiie stroke at -which the steam 


.i :": :r" 




-j: j:r:kr 


































■:: :'_e 


















": "> in 


JL 


JL_ 


1. 


-i- - 


i 


i 


1 


_!_ 


izi-jies. 


3 


24 


4 


" 


. 


: 


" 


" 


_^ 


■;v4 


0"45 


i ■ : :■ 


E 4S 


4-c-: 


4*21 


1,7 


2-43 


2S£ 


6~79 


-:-u 


5-SS 


536 


i - \ 


4-13 


3-35 


_ :. : 


23 


665 


6-21 


: n 


: 24 


4--;.r 


4!7 


3-32 


I M 


:_:- 


■:■■:■:■ 


6-OT 


l -.2 


513 


1 :; 


£v5 


S-2S 


1 _; 


_:" 


636 


5-94 


l-V} 


5-02 


4-4i' 


b-SS 


3-13 


. 24 


21£ 


6-51 


5-SJ 


5^5 


4-:> 


4-SI' 


S-Sj 


3-10 


2-19 


21 


6-07 


:■:- 


1-21 


4-79 


4'25 


: -: 


s-:s 


2-14 


204 


"V- 


5-53 


512 


4-67 


415 


3-63 


298 


■" • " a 


20 


■:•:= 


540 


*•:•:■ 


4'56 


4-:s 


3-54 


i-59 


2*04 


i:~ 


: 54 


- 21 


4:7 


4 - 4-!i 


£vS 


3-45 


1 82 


: ■ : : 


19 


-■^ 


: ll 




4-33 


£SS 


3-36 


2-^4 


1-94 


is- 


5-34 


±-cl 


4- ; - 




c-7 


3-27 


: .- 


1-88 


is 


- ■:: 


■k-'-z- 


■i-c :■ 


4-::- 


I :" 


3-19 


:■•:: 


1-83 


:~ : 


!■-:•:■ 




4"S7 


3-99 


357 


£■:: 


: :: 


1-78 


::' 


4'.rl 


4-59 


4 .: 


£•>> 


£47 


£••:: 


2-45 


1 -\ 


i:-r 


477 


4-45 


4-12 


3-76 


b-cc 


i-?2 


2-38 


: -:5 


16 


±■1-2 


482 


4- : ;■■ 


£■-;' 


_. 


2-83 


2-31 


1-63 


I 154. 


44S 


4-15 


S-S7 


3-53 


c--:-3 




2-24 


I 18 



PROPORTIONS OF LAP FOR EXPANSION. 

table — Continued. 



191 



Length. 




Proportion of the stroke at which the steam 


is ctit off 




of stroke 
of the 






























valve in 


















inches. 


i 

3 


2 4 


i 

4 


5 
24 


t> 


i 

8 


l 

12 


2~4 


15 


4-33 


4-05 


3-75 


3-42 


3-06 


2-65 


2-16 


1-55 


1<H 


4'19 


3 


91 


3-62 


3-31 


2-96 


2-57 


2-09 


1-48 


14 


4-05 


3 


78 


3-50 


3-19 


2-86 


2-48 


2-02 


143 


13| 


3-90 


3 


64 


3-37 


3-08 


2-75 


2-39 


1-95 


1-37 


13 


3*76 


3 


51 


3-25 


2-96 


2-65 


2-30 


1-88 


132 


12i 


3-61 


3 


37 


3-12 


2-85 


2-55 


2-21 


1-80 


1-27 


12 


3-47 


3 


24 


3-00 


2-74 


2 45 


2-12 


>-73 


1-22 


"i 


3-32 


O 


10 


2-87 


2-62 


2-35 


2 : 03 


166 


1-17 


11 


3-18 


2 


97 


2-75 


2-51 


2-24 


1-95 


1-58 


1-12 


10. v 


3-03 


2 


83 


2-62 


2-39 


2-14 


1-86 


1-51 


1-07 


10 


2-89 


2 


70 


2-50 


2-28 


2-04 


1-77 


1-44 


1-02 


9.V 


2-65 


2 


56 


2-37 


2-17 


1-93 


1-68 


1-32 


•96 


9^ 


2-60 


2 


43 


2-25 


2-05 


1-84 


1-59 


1-30 


•92 


3-} 


5-46 


2 


29 


2-12 


1-94 


1-73 


1-50 


1-23 


•86 


8 


2-31 


2 


16 


2-00 


1-82 


1-63 


1-42 


1-15 


•81 


n 


2-16 


2 


02 


1-87 


1*71 


1-53 


1-33 


1-08 


•76 


7 


2-02 


1 


89 


1-75 


1-60 


1-43 


1-24 


1-01 


•71 


GJ- 


1-88 


1 


75 


1-62 


1-48 


1-32 


1-15 


•94 


•66 


6 


1-73 


1 


62 


1-50 


1-37 


1-22 


1-06 


•86 


•61 


H 


1-58 


1 


48 


1-37 


1.25 


1-12 


•97 


•79 


•56 


5 


1-44 


1 


35 


1-25 


1-14 


1-02 


•88 


•72 


•51 


4.; 


1-30 


1 


21 


1-12 


1-03 


•92 


•80 


•65 


•46 


4' 


1-16 


1 


08 


1-00 


•91 


•82 


•71 


•58 


•41 


3.V 


1-01 




94 


•87 


•80 


•71 


•62 


•50 


•35 


3 


•86 


•81 


•75 


•68 


•61 


•53 


■44 


•30 | 



The above table is an extension of the first, for the purpose 
of obviating, in most cases, the necessity of even the very small 
degree of trouble required in multiplying the stroke of the valve 
by one of the decimals in the first table. The first line of the 
second table consists, as in the first table, of eight fractions, in- 
dicating the various parts of the stroke at. which the steam may 
be cut off. The first column on the left hand consists of various 
numbers that represent the different lengths that may be given 
to the stroke of the valve, diminishing by half inches from 24 
inches to 3 inches. Suppose that we wish the steam to be cut 



192 THEOUT OF THE STEAM-EKOINE. 

off at any of the eight parts of tlie stroke indicated in the first 
line of the table (say at | from the end of the stroke), we find 
4 at the top of the 6th column from the left. We next look for 
the proposed length of stroke of the valve (say 17 inches) in the 
first column on the left. From 17, in that column, we run along 
the line towards the right, and in the sixth column, and directly 
under the £ at the top, we find 8-47, which is the amount of lap 
required in inches to cause the steam to he cut off at 4- from the 
end of the stroke, if the valve has no lead. If we wish to give 
it lead (say J inch), we subtract the half of that, or £==-125 inch, 
from 3-47, and we have 3 47—125 =3-345 inches, the quantity 
of lap that the valve should have. 

To find the greatest efficient breadth that we can give to 
the port in this case, we have, as before, half the length of 
stroke, 8| — S'345==5"155 inches, which is the greatest efficient 
breadth we can give to the port with this length of stroke. It 
is scarcely necessary to observe that it is not at all essential that 
the port should be so broad as this ; indeed, where great length 
<: f stroke in the valve is not inconvenient, it is always an advan- 
tage to make it travel farther than is just necessary to make the 
port open fully ; because, when it travels further, both the ex- 
hausting and steam ports are more quickly opened, so as to al- 
low greater freedom of motion to the steam. 

The manner of using this table is so simple, that we need 
not trouble ourselves with more examples, and may pass on, 
therefore, to explain the use of the third table. 

Suppose that the piston of a steam-engine is making its 
downward stroke, that the steam is entering the upper part of 
the cylinder by the upper steam port, and escaping from below 
the piston by the lower exhausting port ; if, as is generally the 
case, the slide-valve has some lap on the steam side, the Tipper 
port will be closed before the piston gets to the bottom of the 
stroke, and the steam above then acts expansively, while the 
communication between the bottom of the cylinder and the con- 
denser still continues open, to allow any vapour from the con- 
densed water in the cylinder, or any leakage past the piston, to 
escape into the condenser ; but, before the piston gets to the 



EFFECTS OF LAP ON EDUCTION. 193 

bottom of the cylinder, this passage to the condenser will also be 
cut off by the valve closing the lower port. Soon after the lower 
port is thus closed, the upper port will be opened towards the 
condenser, so as to allow the steam that has been acting expan- 
sively to escape. Thus, before the piston has completed its 
stroke, the propelling power is removed from behind it, and a 
resisting power is opposed before it, arising from the vapour in 
the cylinder, which has no longer any passage open to the con- 
denser. It is evident, that if there is no lap on the exhausting 
side of the valve, the exhausting port before the piston will be 
closed, and the one behind it opened, at the same time ; but, if 
there is any lap on the exhausting side, the port before the pis- 
ton will be closed before that behind it is opened ; and the in- 
terval between the closing of the one and the opening of the 
other, will depend on the quantity of lap on the exhausting side 
of the valve. Again, the position of the piston in the cylinder, 
when these ports are closed and opened respectively, will depend 
on the quantity of lap that the valve has on the steam side. If 
the lap is large enough to cut the steam off when the piston is 
yet a considerable distance from the end of its stroke, these 
ports will be closed and opened at a proportionably early 
part of the stroke ; and in the case of engines moving at 
a high speed, it has been found that great benefit is obtained 
from allowing the steam to escape before the end of the 
stroke. 

The ihird table is intended to show the parts of the stroke 
where, under any given arrangement of slide valve, the eduction 
ports close and open respectively, so that thereby the engineer 
may be able to estimate how much, if any, of the efficiency he 
loses, while he is trying to add to the power of the steam by in- 
creasing the expansion. In this table there are eight columns 
marked A, standing over eight columns marked B, and at the 
heads of these columns are eight fractions as before, representing 
so many different parts of the stroke at which the steam may 
be supposed to be cut off. 



194 



THEORY OF THE STEAM-ENGINE. 



The columns marked A express the distance of the piston-* 
in parts of it 3 stroke — from the end of the stroke ichen the educ- 
tion port oefore it is shut, and the columns marked B, and which 
stand immediately under the columns marked A, express thi 
distance of the piston from the end of its stroke ic7ien the ex- 
hausting port oehind it is opened — also measured in parts of the 
stroke.* 

m. PEOPOETION OF THE STROKE AT WHICH THE EDUCTION 

POET IS SHUT AND CPEXED. 



Lap on the 
eduction side of the 




Proportion of the stroke at which the steaai is 


rat off 




valve, in part; of 
the length of its 


































stroke. 


i 


■4 


1 
•4 


5 

1-t 


i 


£ 


11 


24 




A 


A 


A 


A 


A 


A 


A 


A 


1-Sth 


•ITS 


■161 


•143 


•126 


•109 


•093 


■074 


•053 


l-16th 


•130 


•113 


•100 


•085 


•071 


•058 


•043 


•027 ' 


l-32nd 


•118 


•101 


•085 


•069 


•053 


•043 


•033 


•024 1 





•092 


•082 


•067 


•055 


•043 


•033 


•022 


•011 




B 


B 


B 


B 


B 


B 


B 


B 


1-Sth 


•033 


•026 


•019 


•012 


•oos 


•004 


•001 


•001 


l-16th 


•060 


•052 


•040 


•030 


•022 


•015 


•oos 


•002 


l-32nd 


•073 


•066 


•051 


•042 


•033 


•023 


•013 


•004 





■092 


■082 


•067 


•055 


•044 


•033 


•022 


•011 



Suppose we have an engine in which the slide valve is made 
to cut the stem off when the piston is l-3rd from the end of its 



* In locomotive and other fast-moving engines it is very important to open the 
eduction passage before the end of the stroke, so as to give more time for the 
steam to escape, and in locomotive valves the lap of the valve is usually made a 
little over J th of the travel, and the lead is usually made T \th of the travel. In 
engines moving slowly the same necessity for an early eduction does not exist, and 
in such engines there will be a loss from opening the eduction much before the end 
of the stroke, as the moving pressure urging the piston is thus removed before the 
stroke terminates. When the valve is closed before the piston previously to the 
end of the stroke, the attenuated vapour in the cylinder will be compressed, and 
sometimes the compression will be carried so far that the pressure resisting the 
piston at the end of the stroke will exceed the pressure of the steam in the. boiler. 
The indicator diagram will in such cases appear with a loop at its upper corner, 
which shows that the pressure before the end of the stroke exceeds the pressure of 
the steam, and that the first effect of opening the communication between t&e 
cylinder and the boiler is to enable the cylinder to discharge its highly compressed 
vapour backward into the boiler. The act of compressing the steam is what is 
called cushioning and in all ordinary diagrams this action may bo more or loes 
pcrcci ved. 



EFFECTS OF LAP ON EDUCATION. 195 

stroke, and that the lap on the eduction or exhausting side of 
the valve is l-8th of the "whole length of its stroke. Let the 
stroke of the piston be 6 feet, or 72 inches. "We wish to know 
when the exhausting port before the piston will be closed, and 
when the one behind it will be opened. At the top of the left- 
hand column marked A, the given degree of expansion (l-3rd) 
is given, and in the extreme left column we have at the top the 
given amount of lap (l-8th). Opposite the l-8th in the imt 
column, marked A, we have *178, and in the first column, marked 
B, -033, which decimals, multiplied respectively by 72, the length 
of the stroke, will give the required positions of the piston: 
thus 72 x '178 = 12*8 inches = distance of the piston from the 
end of the stroke when the exhaustion-port before the piston is 
shut: and 72 x - 033 = 2*38 inches = distance of the piston from 
the end of its stroke when the exhausting-port behind it is 
opened. 

To take another example. Let the stroke of the valve be 16 
inches, the lap on the exhausting side \ inch, the lap on the 
steam side 3 \ inches, and the length of the stroke of the piston 
60 inches. It is required to ascertain all the particulars of the 
working of this valve. The lap on the exhausting side is evi- 
dently ^ of the length of the valve stroke. Then, looking at 16 
in the left-hand column of the table in page 190, we find in the 
same horizontal line, 3*26, or very nearly 3£, under \ at the head 
of the column, thus showing that the steam will be cut off at 
one-sixth from the end of the stroke. Again, under \ at the 
head of the sixth column from the left in the table in page 194, 
and in a line with ^ in the left-hand column, Ave have *053 un- 
der A, and '033 under B. Hence, -053 x 60 = 3*18 inches = dis- 
tance of the piston from the end of its stroke when the exhaust- 
ing-port before it is shut, and *033 x 60 "= 1*98 inches = distanco 
of the piston from the end of its stroke when the exhausting- 
port behind it is opened. If in this valve the lap on the ex- 
hausting side were increased say to 2 inches or \ of the stroke, 
the effect would be to cause the port before the valve to be shut 
sooner in the proportion of *109 to "053, and the port behind it 
later in the proportion of *008 to *003. "Whereas, if the lap on 



196 THEORY OF THE STEAM-ENGINE. 

the exhausting side were removed entirely, the port before th€ 
piston would be shut and that behind it opened at the same 
time. The distance of the piston from the end of its stroke at 
that time would be '043 x 60 = 2*58 inches. 

An inspection of the third table shows ns the effect of in- 
creasing the expansion by the slide "valve in augmenting the losa 
of power occasioned by the imperfect action of the eduction pas- 
sages. Eeferring to the bottom line of the table, we see that the 
eduction passage before the piston is closed, and that behind i' 
opened, thus destroying the whole moving power of the engine, 
when the piston is '092 from the end of its stroke, the steam 
b3ing cut off at •£ from the end. "Whereas if the steam is only 
cut off at 2j from the end of the stroke, the moving power is not 
withdrawn till only -Oil of the stroke remains uncompleted. It 
will also be observed that increasing the lap on the exhausting 
side has the effect of retaining the action of the steam longer 
"behind the piston, but it at the same time causes the eduction 
port "before it to be closed sooner. 

A very cursory examination of the action of the slide valve 
is sufficient to show that the lap on the steam side should always 
be greater than on the eduction side. If they were equal, the 
steam would be admitted on one side of the piston at the same 
time that it was allowed to escape from the other ; but universal 
experience has shown that when this is the case a very con- 
siderable part of the power of the engine is destroyed by the re- 
sistance opposed to the piston, by the escaping steam not getting 
away to the condenser with sufficient rapidity. Hence we see 
the necessity of the lap on the eduction side being always less 
than the lap on the steam side ; and the difference should be the 
greater the higher the velocity of the piston is intended to be, 
because the quicker the piston moves, the passage for the waste 
steam requires to be the larger, so as to admit of its getting away 
to the condenser with as great rapidity as possible. In locomo- 
tive or other engines, where it is not wished to expand the steam 
in the cylinder at all, the slide valve is sometimes made with 
revy little lap on the steam side ; and in these circumstances, in 
. mier to get a sufficient difference between the lap on the steam 



EFFECTS OF LAF ON EDUCTION. 197 

and the eduction sides of the valve, it niay be necessary not only 
to take away all the lap on the eduction side, but to take off still 
more, so as to cause both eduction passages to be, in some de ■ 
gree, open, when the valve is at the middle of its stroke. This, 
accordingly, is sometimes done in such circumstances as we have 
described ; but, when there is a considerable amount of lap on 
the steam side, this plan of taking more than all the lap off tt e 
eduction side ought never to be resorted to, as it can serve no 
good purpose, and will materially increase an evil we have al- 
ready explained : viz., the opening of the eduction port behind 
the piston before the stroke is nearly completed. In the case 
of locomotive or other engines moving rapidly, it is very con- 
ducive to efficiency to begin, the eduction before the end of the 
stroke, as the piston moves slowly at that time ; and a very small 
amount of travel in the piston at that point corresponds to a 
considerable additional time given for the accomplishment of the 
eduction. The tables apply equally to the common short-slide 
three-ported valves, and to the long D valves. 

The extent to which expansion can be carried conveniently 
by means of lap upon the valve is about one-third of the stroke ; 
that is, the valve may be made with so much lap that the steam 
will be cut off when one-third of the stroke has been performed, 
leaving the residue to be accomplished by the agency of the ex- 
panding steam ; but if much more lap be put on than answers to 
this amount of expansion a distorted action of the valve will be 
produced, which will impair the efficiency of the engine. By 
the use of the link motion, however, much of this distorted action 
can be compensated. If a farther amount of expansion than this 
is wanted, where the link motion is not used, it may be attained 
by wire-drawing the steam, or by so contracting the steam pas- 
sage that the pressure within, the cylinder must decline when the 
speed of the piston is accelerated, as it is about the middle of the 
stroke. Thus, for example, if the valve be so made as to shut 
off the steam by the time two- thirds of the stroke have been 
performed, and the steam be at the same time throttled in the 
steam pipe, the full pressure of the steam within the cylinder 
cannot be maintained except near the beginning of the stroke, 



198 THEOET OF THE STEA3I-ENGESE. 

wheie the piston travels slowly; for as the speed of the piston 
increases, the pressure necessarily subsides, until the piston ap- 
proaches the other end of the cylinder, where the pressure would 
rise again hut that the operation of the lap on the valve by this 
time has had the effect of closing the communication between 
the cylinder and steam pipe, so as to prevent more steam from 
entering. By throttling the steam, therefore, in the manner 
here indicated, the amount of expansion due to the lap may be 
doubled, so that an engine with lap enough upon the valve to 
cut off the steam at two-thirds of the stroke, may, by the aid 
of wire-drawing, be virtually rendered capable of cutting off the 
steam at one-third of the stroke. 

The Link Motion. — The rules and proportions here given, 
are equally applicable, whether the valve is moved by a single 
eccentric, or by the arrangement called the link motion, and 
which has now been very generally introduced into steam en- 
gines. In the link motion there are two eccentrics, one of 
which is set so as to drive the engine in one direction, and the 
other is set so as to drive the engine in the opposite direction, 
and when the stud in communication with the valve is shifted to 
one end of the link, that stud partakes of the motion of the for- 
ward eccentric, whereas, when it is placed at the other end of 
the link, it partakes of the motion of the backing eccentric. A 
common length of the link is three times the stroke of the valve. 
Generally the stud is placed either at one end of the link or the 
other, not by moving the stud but by moving up or down the 
link ; and it is better that this movement should be vertical, and 
be made by means of a screw, than that the movement should 
be produced by a lever travelling through an arc. The point 
of suspension should be near the middle of the link where its 
motion is the least. The link connects together the ends of the 
two eccentric rods, and is sometimes made straight, but gen- 
erally curved, the curvature being an arc of such radius that the 
link may be raised up or down without sensibly altering the 
position of the stud with which the valve is connected. But the 
jink should be convex or concave towards the valve, according 
as tie eccentric rods are crossed or uncrossed when the throw 



VELOCITY OF RUNNING WATER IN CONDUITS. 199 

of the eccentrics are turned towards the link. In the case of 
new arrangements of engine, it is advisable to make a skeleton 
model in paper of the link and its connexions, so as to obtain full 
assurance that it works in the best way. 

VELOCITY OF WATER IX RIVERS, CANALS, AND PIPES, 
ANSWERABLE TO ANY GIVEN DECLIVITY. 

When a river runs in its bed with a uniform velocity, the 
gravitation of the water down the inclined plane of the bed, is 
just balanced by the friction. In the case of canals, culverts, 
and pipes, precisely the same action takes place. The head of 
water, therefore, which urges the flow through a pipe, may be 
divided into two parts, of which one part is expended in giving 
to the water its velocity, and the other part is expended in over- 
coming the friction. If water be let down an inclined shoot, its 
motion at the top will be slow, but will go on accelerating until 
the friction generated by the high velocity will just balance the 
gravitation down the plane, and after this point has been attained, 
the shoot may be made longer and longer without any increase 
in the velocity of the water taking place. In the case of a ball 
falling in the air or in. water, the velocity of the descent will go 
on increasing until the resistance becomes so great as to balance 
the weight ; and, in the case of a steam vessel propelled through 
the water, the speed will go on increasing until the resistance 
just balances the tractive force exerted by the engines, when the 
speed of the vessel will become uniform. In all these cases the 
resistance increases with the speed ; and as the speed increases, 
the resistance increases also, until it becomes equal to the ac- 
celerating force. 

The resistance which is occasioned by the friction of water 
increases more rapidly than the increase of the velocity. In 
other words, there will be more than twice the friction with 
twice the velocity. It is found by experiment that the friction 
of water increases nearly as the square of its velocity, so that 
there will be about four times the resistance with twice the 
speed. This law, however, is only approximately correct. The 



200 THEORY OF THE STEAM-ENGINE . 

friction does not increase quite so rapidly at high velocities ai» 
the square of the speed. 

It is easy to determine the friction in lbs. per square foot of 
any given pipe or conduit, -with any given velocity of the stream, 
when the slope or declivity of the surface of the -water is known. 
For as the gravitation down the inclined plane of the conduil 
just balances the friction, the friction in the whole length of the 
conduit will he equal to the whole weight of the water in it, rev 
duced in the same proportion as any other body descending an 
inclined plane. Thus, if the conduit he 2,000 feet long, and have 
1 foot of fall in that length, the total friction will he equal to the 
total weight of the water divided by 2,000, and the friction per 
square foot will be equal to this 2000th part of the weight of the 
water divided by the number of square feet exposed to the water 
in the conduit. The friction will in ail cases vary as the rubbing 
surface, or, what is the same thing, as the wetted perimeter 
As a cylindrical pipe has a less perimeter than any other form, it 
will occasion less resistance than any other form to water passing 
through it. In like manner, a canal or a ship with a semi- 
circular cross section will have the minimum amount of friction. 

The propelling power of flowing water being gravity, the 
amount of such power will vary with the magnitude of the 
stream ; but the resisting power being friction, which varies with 
the amount of surface, or in any given length with the wetted 
perimeter, it will follow that the larger the area is relatively 
with the wetted perimeter, the less will be the resistance rela- 
tively with the propelling power, and the greater will be the 
velocity of the water with any given declivity. Xow, as the 
circumference or perimeter of a pipe increases as the diameter, 
and the area as the square of the diameter, it is clear, that with 
any given head, water will run more swiftly through large pipes 
than through small ; and in like manner with any given propor- 
tion of power to sectional area, large vessels will pass more 
swiftly than small vessels through the water. The sectional 
area of a pipe or canal divided by the wetted perimeter, is what 
is termed the hydraulic mean dejyth, and this depth is what 
i* Duld result if we suppose the perimeter to be bent out to a 



VELOCITY OF RUNNING WATER IN CONDUITS. 201 

straight line, and the sectional area to be spread evenly over it, 
so that each foot of the perimeter had its proper share of sec- 
tional area above it. The greater the hydraulic mean depth, the 
greater -with any given declivity will be the velocity of the 
stream. With any given fall, therefore, deep and large rivers 
will run more swiftly than small and shallow ones. The hy- 
draulic mean depth of a steam vessel will be the indicated power 
divided by the wetted perimeter of the cross section. 

TO DETERMINE THE MEAN VELOCITY WITH WHICH WATER WILL 
FLOW THEOUGH CANALS, AETEEIAL DRAINS, OR PIPES, RUN- 
NING PARTLY OR WHOLLY FILLED. 

Ettle. — Multiply the hydraulic mean depth in feet by twice the 
fall in feet per mile ; talce the square root of the product and 
multiply it by 55. Th-e result is the mean 'velocity of the 
stream in feet per minute. This again multiplied by the sec- 
tional area in square feet gives the discharge in cubic feet 
per minute. 

Example.— What is the mean velocity of a river tailing a foot 
in the mile, and of which the mean hydraulic depth is 8 feet ? 

Here 8 x 2 = 16, the square root of which is 4, and this 
multiplied by 55 = 220, which will be the mean velocity of the 
stream in feet per minute. 

In cylindrical pipes running full, the hydraulic mean depth is 
one-fourth of the diameter. For the hydraulic mean depth being 

the area divided by the wetted perinieter.it is = - — * i 

1 ' 3-1416^ 4* ; 

* The surface, bottom, and mean velocities of rivers have fixed relations to ono 
another. Thus, if the surface velocity in inches per second be denoted by V, thft 
mean velocity will be (Y + - 5) — \/V and the bottom velocity by (V + 1) — 2 y'Y. 
With surface velocities therefore of 4,16, 32, 64, and 100 inches per second, the cor- 
responding mean velocities will be 2 - 5, 12'5, 26*8, 56'5, and 90*5 inches per second, 
and the corresponding bottom velocities will be 1, 9, 21 - 6, 49, and 81 inches per 
second. 

The common rule for finding the number of cubic feet of water delivered each 
minute by a pipe of any given diameter is as follows : — Divide 4 - 72 times the square 
root of the fifth power of the diameter of the pipe in inches by the square root 
of the quotient obtained by dividing the length of the pipe in feet by the head 
of water in feet. Hawksley's rule for ascertaining the delivery in gallons per 
hour is as follows : — Multiply 15 times the fifth potcer of the diameter of the pipe 

9* 



202 THEORY OF THE STEAM-ENGINE. 

M. Prony has shown by a comparison of a large number of 
experiments that if H be the head in feet per mile required to 
balance the friction, Y the Telocity of the water through the pipa 
in feet per second, and D the diameter of the pipe in feet, then 

• 2-25Y- 

This equation is identical with that which has been used by 
Boulton and Watt in their practice for the last half century, and 
which is as follows : — 

If I be the length of the main in miles, Y the velocity of the 

water in the main in feet per second, D the diameter of the pipe 

in feet, and 2*25 a constant, 

2-25ZY 2 
then — jz — = feet of head cine to friction. 

This equation put into words gives us the following Rule:— 

TO DETEEMIXE THE HEAD OF WATEE THAT WILL BALANCE THE 
FEICTION OF WATEE EUKNUJTG WITH ANY GIVEN VELOCITY 
THROUGH A PIPE OF A GIVEN LENGTH AND DIAMETER. 

Rule. — Multiply 2*25 times the length of the pipe in miles oy 

the square of the velocity of the water in the pipe in feet per 

second, and divide the product oy the diameter of the pipe in 

feet. The Quotient is the head of water in feet that will 

oalance the friction. 

The law indicated by. this Rule is expressed numerically in 
the Tables on pp. 204, 205. 

in inches by the head of water in feet, and divide the 'product by the length of 
the pipe in yards. Finally, extract the square root of the quotient, ichich givet 
the delivery in gallons per hour. 

The annual rain-fall in England varies from 20 to TO inches, the mean being 42 
Inches, and it is reckoned that about jgths of the rain-fall en any given area may 
be collected for storage. A cubic foot of water is about 6| gallons, and it is found 
n supplying towns with water that about on the average 16 gallons per head per 
3ay are required in ordinary towns, and 20 gallons per head per day in manufac- 
turing towns, but the pipes should be large enough to convey twice this quantity. 
7n the rainy districts of England collecting reservoirs should contain 120 days 1 sup- 
ply, and in dry districts 200 days" supply. Service reservoirs are usnally made to 
contain 3 days' supply. The mean daily evaporation in England is "03 of an inch, 
and the loss from the overflow of storm water is reckoned to be about 1 per cent. 



FRICTION AND DISCHARGE OF WATER. 203 

Explanation of the Tables.— The top horizontal row of 
figures represents cither the diameter of a cylindrical pipe, or 
four times the area of any other shaped pipe divided by the cir- 
cumference, or four times the area of the cross section of a canal, 
divided by the sum of all its sides, or bottom and sides, all being 
in inches. 

The first vertical column indicates the slope of the pipe or 
canal, that is, the whole length of the pipe or canal, divided by 
the perpendicular fall. 

Any number in any other column indicates the velocity, in 
inches per second, with which water would run through a pipe 
of such a diameter as the number at the head of such column 
expresses, having such a slope as that number in the first column 
expresses which is horizontally against such velocity. 

Example 1. — With what velocity will water run through a 
pipe of 16 inches diameter, its length being 8,000 feet, and fall 
16 feet? Here the slope manifestly is 8,000-5-16=500. Against 
500 in the first column, and under 16, the diameter in the top 
row of figures, the number 29*8 is found, which is the velocity 
in inches per second. 

Example 2. — With what velocity will water pass through a 
pipe of 21 inches diameter, having a slope of 900 ? 21 is not 
found in the head of the Table, in which case such a number 
must be found in the top row as will bear such proportion to 21 
as some other two numbers in the top row bear to each other, 
and these latter numbers should be as near to 21 as they can be 
found. 

In this case it will be seen that 18 is to 21 as 6 is to 7, or 
(for compliance with the indication just mentioned) rather as 12 
to 14, or still better as 24 to 28. Then say as the velocity 
(against 900, the slope) under 24 is to 28 (28*7), so is the velocity 
under 18 (22*7) to that of 21 (viz. 24*7) the velocity in inches 
per second. 

By the same process may the velocity for slopes be found or 
assigned, which are not to be found in the first column of tho 
Table, proceeding with proportions found in the vertical col- 
umn instead of the horizontal rows ; the first vertical column 



204 



THEORY OF THE STEAM-EXGESTE. 



VELOCITY IN INCHES PEE SECOND OF WATEB FLOWING Til ROUGH 
PIPES WITH YABIOT73 SLOPES AND DIAMETEES. 

BY BOULTOS, WATT & CO. 



1 

Slope or 
( Length 
1 di yided by 
Fall. 


INTERNAL DIAMETERS OF THE PIPES EN - INCHES. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


12 


14 


10 


63"3 


96-0 


121- 


142- 


161- 


ISO- 


193- 


208- 


221* 


234- 


255- 


230 


20 


418 


63-4 


80- 


93-6 


104- 


113- 


127- 


137- 


146- 


154- 


170- 


185- 


30 


S2-3 


49-5 


62-G 


734 


83'3 


931 


99-7 


107' 


114- 


121- 


133- 


145- 


40 


27'5 


41-5 


52-5 


61-5 


69-8 


7S-0 


83-7 


901 


95-8 


103- 


112- 


121- 


50 


23-9 


36'3 


457 


537 


60"9 


68'0 


73'Q 


787 


835 


88'4 


97'5 


106" 


60 


21-6 


32-7 


41"2 


4S-3 


54-8 


61-2 


65-7 


70-8 


75-3 


79-8 


87-8 


954 


70 


19-6 


29'7 


37-5 


44-0 


49-8 


55-7 


59-7 


64-4 


68-5 


72*5 


80-0 


866 


80 


181 


27-4 


34-7 


40-7 


461 


51-5 


553 


39-5 


633 


671 


74-0 


S0-2 


90 


16-9 


25-6 


324: 


37'9 


43-0 


43-0 


5T5 


55'5 


590 


62-5 


69-0 


74-3 


100 


15'8 


240 


30'3 


35'5 


40'2 


45"1 


48'4 


522 


555 


587 


64'8 


703 


200 


10-5 


16-0 


20-2 


23-7 


26-8 


30-0 


322 


347 


3GD 


39-0 


431 


46-7 


800 


8-43 


12-7 


161 


18-6 


21-4 


23-8 


25-6 


27-7 


29-4 


30-6 


34-3 


37-7 


400 


7-11 


10-3 


13 6 


15-9 


1S1 


20-2 


21-6 


23-3 


24-8 


26-8 


29-0 


81-4 


500 


628 


9"50 


12-0 


14'0 


15'9 


17'8 


191 


206 


21'9 


23'2 


255 


277 


600 


564 


8-57 


10-3 


12-7 


14-3 


161 


17-2 


18'6 


19-7 


20-9 


23-0 


25-0 


TOO 


517 


7-85 


9-90 


11-6 


13-2 


14-7 


15-8 


170 


13-1 


19-2 


211 


22-9 


800 


4-S1 


7-30 


9-21 


10-3 


12-2 


13-7 


14-7 


15-8 


16-8 


17-8 


19-6 


21-3 


900 


4-50 


6-83 


8-62 


101 


11-4 


12-3 


13-7 


14-3 


15-7 


16-6 


18-3 


19-9 


1000 


4"25 


6'45 


8*15 


9'54 10'S 


12*1 


12'9 


140 


14'8 


157 


17'3 


18-8 


2000 


2-83 


4-37 


5-52 


643 


7-33 


S-2 


8-77 


9-4S 


101 


10-6 


11-7 


12-7 


3000 


2-30 


3-43 


4-40 


517 


5-86 


6-55 


7-02 


7 - 57 


S-05 


8-52 


940 


10-2 


4000 


1-96 


2-97 


3-75 


4-40 4-98 


5-57 


5-97 


6-44 


6'84 


7-25 


8-00 


8-66 


5000 


173 


2'62 


331 


3-38 4'40 


4*92 


5 28 5 69 604 


640 


7 06 


7'66 


6000 


1-5T 


2-33 


3-00 


3-52 


399 


4-45 


4-79 


515 5-44 


5-SO 


6-40 


6-95 


7000 


1-43 


217 


2-73 


3-21 


3-63 


4-06 


4-36 


4-70 5 00 


5-29 


5-82 


6-32 


8000 


1-32 


2-01 


2-53 


2-87 


3-46; 3-76 


4-03 


4-35 4-62 


4-90 


540 


5S5 


9000 


1-24 


1-87 


2-33 


2-79 3-16| 8-53 


8-79 4-08J 4-34 4-60 


5-07 


5-50 


10000 


1-17 


177 


2'24 


2 62 29 1 3*32 


357 3'84| 408 4*32 


476 


516 



being substituted in this case for the top row in tlie former 
case. 

In all cases an addition must be made to the fall equal to that 
which would generate the existing velocity in a body falling 
freely by gravity. For instance, in the first case, to the fall of 



VELOCITY OF WATEll IN PIPES. 



205 



VELOCITY IX INCHES PER SECOND OF WATER FLOWING THROUGH 

pipes with yarious slopes and diameters. — {Continued.) 

BY BOULTOK, WATT & CO. 



Slope or 

Length 

divided *>y 

Fall. 


INTERNAL DIAMETERS OF THE PIPES IN INCHES. 


16 


18 


29 


24 


28 


82 


36 


40 


60 


80 


1 30 

775- 


10 


30P 


320- 


338- 


372- 


403- 


432- 


459- 


484- 


597- 


692- 


20 


199- 


21P 


223* 


245- 


256- 


2S5- 


803- 


320- 


394- 


456- 


511- 


30 


155' 


165- 


175- 


192- 


203- 


223- 


238- 


250- 


309- 


358- 


400- 


40 


130" 


13S- 


146- 


161- 


174- 


1ST* 


199- 


210- 


259- 


300- 


336- 


50 


113* 


121" 


127* 


140" 


152* 


163* 


173' 


183* 


226* 


261* 


293' 


60 


102- 


109- 


115- 


127- 


137- 


147* 


156- 


165- 


204- 


236- 


264- 


70 


93-2 


99-0 


104- 


115- 


124- 


134- 


142- 


150- 


135- 


214- 


240- 


80 


86-3 


91-6 


97' 


106- 


115- 


123- 


ISP 


139- 


17P 


198- 


222- 


90 


80-5 


85-5 


90-4 


99-4 


107- 


115- 


122- 


129- 


159- 


185- 


207- 


100 


75*5 


80'2 


85*0 


93'5 


102* 


108' 


115' 


121* 


150* 


173* 


194' 


200 


50-2 


53-3 


56-4 


62-0 


67-2 


72-1 


76-7 


80-8 


99-5 


115- 


129- 


300 


39-9 


42-5 


45-0 


49-5 


53-6 


57-5 


61-1 


64-4 


79-5 


92-0 


105- 


400 


33-8 


35-9 


3S-0 


41-8 


45-2 


48-5 


51-5 


54-4 


67-0 


77-7 


87-0 


500 


29'8 


31-6 


33"4 


36'8 


39*8 


42'8 


45*5 


47 9 


590 


68'4 


767 


600 


26-8 


28-6 


30-2 


33-2 


36-0 


38-6 


41-0 


43-2 


53-3 


61-7 


69-2 


700 


24-6 


26-2 


27-7 


30-4 


33-0 


35-3 


37*6 


39-6 


48-8 


56-5 


63-4 


800 


22-9 


24-3 


25-7 


28-3 


30-6 


32-9 


34-9 


36-8 


45-4 


525 


58-9 


900 


21-4 


22-7 


24-0 


26-4 


28-7 


30-7 


32-7 


34-4 


42-5 


491 


55-1 


1000 


20"2 


21*5 


22'7 


25*0 


27*1 


29*1 


309 


325 


40-1 


46'4 


520 


2000 


13-T 


14-5 


15-4 


16-9 


18-3 


19-7 


20-9 


22-0 


27-2 


31-3 


35-3 


3000 


10-9 


11-6 


12-3 


13-5 


14-6 


15-7 


167 


17-6 


21-7 


25-2 


28-2 


4000 


9-32 


9-90 


10-4 


11-5 


12-4 


13-4 


14-2 


15-0 


1S-5 


21-4 


24-0 


5000 


8*23 


873 


925 


103 


11-0 


11*8 


12/5 


13'2 


163 


18'9 


21*2 


6000 


7-47 


7-93 


8-40 


9-23 


10-0 


10-7 


11-4 


12-0 


148 


174 


19-2 


7000 


6-80 


7-22 


7-65 


8-40 


9-10 


9-76 


10-3 


10-9 


135 


15-6 


17-5 


8000 


6-30 


669 


7-07 


7-78 


8-43 


9-05 


9-62 


10-1 


12-5 


145 


10-2 


9000 


5-91 


6-28 


6-64 


7-30 


7-92 


8-50 


9-02 


9 52 


11-7 


13-05 


15-2 


10000 


5*55 


591 


625 


6*87 


7 '45 


7'98 


8-50 


893 


110 


127 


14'3 



16 feet we mnst add the fall which, would generate the velocity 
of 29*8 inches per second, namely, 1*15 inches, which will make 
the total fall 16 feet 1*15 inches that will he requisite to give 
such a velocity; hut in such cases as this it is evident that tho 
addition of this small fraction might have heen disregarded. 



206 THEORY OF THE STEAM-ENGINE. 

In some cases Messrs. Boulton and "Watt have employed the 
constant 1*82 instead of 2*25. Mr. Mylne's constant is 1:94; 
but some careful experiments made by Mm at the West Middle- 
sex Waterworks, gave a constant as high as 2 "62. 

OTHER TOPICS OF THE THEOET OF STEAAI-ENGEXES. 

It will not be necessary to extend these remarks by an inves- 
tigation of the theory of the crank as an instrument for convert- 
ing rectilinear into rotatory motion, since the idea, once widely - 
prevalent, that there was a loss of power consequent upon it3 
use, is now universally exploded. Neither will it be necessary 
to enter into any explanation of the structure of the numerous 
rotatory engines which have at different times been projected, 
since none of those engines are in common or beneficial opera- 
tion. The proper dimensions of the cold water and feed pumps, 
the action of the fly-wheel in redressing irregularities of the mo- 
tion of the engine, and other material points which might prop- 
erly fail to be discussed under the head of the Theory of the 
Steam-Engine, and which have not already been treated of, 
will, for the sake of greater conciseness, be disposed of in the 
chapter on the Proportions of Steam-Engines, when these vari- 
ous topics must necessarily be considered. Xor is it deemed ad- 
visable here to recapitulate the rules for proportioning the vari- 
ous kinds of parallel motion, since parallel motions have now 
almost gone out of use, and since also any particular case of a 
parallel motion which has to be considered, can easily be resolv- 
ed geometrically by drawing the parts on a convenient scale, — 
the principle of all parallel motions being that the versed sine 
of an arc, pointing in one direction, shall be compensated by an 
equal versed sine of an arc pointing in the opposite direction ; and 
the effect of these opposite motions is to produce a straight hue. 
In the case of the parallel motions sometimes employed in side- 
iever engines, and in which the attachment is made not to the 
cross-head but to the side-rod, it is only necessary to provide 
that the end of the bar connected to the side-rod shall move, 
not in a straight line, but in an arc, the versed sine of which ia 
equal to the versed sine of the arc described by the point of at- 



MODE OF DRAWING THE PARALLEL MOTION. 207 

tachnient on the side-rod. As the bottom of the side-rod is at- 
tached to the beam and the top to the cross-head, and as the 
bottom moves in an arc and the top in a straight line, it is clear 
that every intermediate point of the side-rod must describe an 
arc which will more and more approach to a straight line, or 
have a smaller and smaller versed sine, the nearer such point is 
to the top of the rod. By drawing down the side-rod at the end 
of the stroke, and also at half stroke, the amount of deviation 
from the vertical at those positions, can easily be determined for 
any point in the length of the rod ; and the point of attachment 
of the parallel bar has only to be such, and the length and travel 
of the radius crank has also to be such, that the end of the paral- 
'el bar attached to the side-rod shall describe an arc whose versed 
sine is equal to the deviation from the perpendicular, or, in other 
words, to the side-travel of that point of the side-rod at which 
the attachment is made. Since, then, the side-rod is guided at 
the bottom by the arc of the beam, and near the top by that less 
arc described by the end of the parallel bar, which answers to 
the supposition of the cross-head moving in a vertical line, the 
result is that the cross-head will be constrained to move in this 
vertical line ; since only on that supposition can the two arcs 
already fixed be described. 

The method of balancing the momentum of the moving parts 
of marine engines which I introduced in 1852 has now been 
very generally adopted ; and the practice is found to be very 
useful in reducing the tremor and uneasy movements to which 
engines working at a high rate of speed are otherwise subject. 
Nearly all the engines now employed for driving the screw pro- 
peller are direct-acting engines, which necessarily work at a high 
rate of speed to give the requisite velocity of rotation to the 
screw shaft. The principle on which the balancing is effected is 
that of applying a weight to the crank or shaft, and when the 
piston and its connexions move in one direction the weight 
moves in the opposite with an equal momentum. 



CHAPTER IV. 

PROPORTIONS OF STEAM-ENGINES. 

"We now come to the question how we are to determine the 
proportions of steam-engines of every class. 

The nominal power of a low pressure engine is determined 
by the diameter of the cylinder and length of the stroke, as 
follows : — 



TO DETEEMINE THE NOMINAL POWEE OF A LOW PEESSCEE ENGINE 
OF watt's CONSTETJOTION. 

Rule. — Multiply the square of the diameter of the cylinder in 
inches ~by the cube root of the stroke in feet, and divide the 
product by 47. The quotient is the nominal liorse-power of 
the engine. 

Example 1. — What is the nominal, power of a low pressure 
engine with a cylinder 64 inches diameter and 8-feet stroke ? 

Here 64 x 64 = 4,096, which multiplied by 2, the cube root 
of 8 = 8,192 and -j- 47 = 174-3. 

The nominal powers of engines of different sizes, both 
high pressure and low pressure, are given in the following 
tables : — 



TABLES OF NOMINAL POWERS OF ENGINE. 



209 



NOMINAL IIOESE POWER OF IIIGH PEE3SITEE ENGINES. 



"S.3 

u <- <£ 
* B u 








LENGTH 


OF STROKE IN FEET. 




i 


1* 

•00 


2 


H 


3 


8* 


4 


5 


6 


1 


8 
•51 


9 

•51 


2 


•25 


•32 


•35 


•37 


•38 


•40 


•44 


•46 


•49 


2 * 


•39 


•45 


•50 


•54 


•57 


•60 


•63 


•68 


•72 


•76 


•79 


•83 


O 


•57 


•65 


•72 


•78 


83 


•87 


•91 


•98 


1-04 


M0 


115 


1-20 


S} 


•78 


•89 


•98 


1-06 


1*13 


1*19 


1*24 


1-34 


1-42 


1-49 


1-56 


1-62 


4 


1-02 


1-17 


1*29 


1-38 


1-47 


1-56 


1*62 


1-74 


1-86 


1-95 


2-04 


2-10 


H 


1-29 


1-48 


1*63 


1-75 


1-86 


1*96 


2-05 


2-21 


2 35 


2-47 


2 58 


2-GS 


5 


1-59 


1-S3 


2*01 


216 


2-28 


243 


2-52 


2-73 


2-88 


3-06 


318 


8 33 


5} 


1-93 


2-21 


2-43 


2-62 


2-7S 


2-93 


8-12 


3-30 


3-51 


369 


386 


4-01 


6 


2-28 


2-61 


2-8S 


3-12 


3-30 


8-48 


3-66 


8-93 


4-17 


4-41 


4-59 


4-77 


0} 


2-69 


3-09 


3-39 


3-66 


3-90 


4-08 


4-23 


4-62 


4-89 


5-16 


546 


5-61 


7 


8-12 


3-57 


3-93 


4-23 


4-50 


4-74 


4-95 


5-34 


5-67 


5-97 


6-27 


6-51 


7* 


3-60 


4-11 


4-53 


4-86 


5-19 


5-46 


5-70 


615 


6-51 


6-87 


718 


746 


8 


4-0S 


4-68 


5-16 


5-55 


5-88 


6*21 


6-4S 


6-99 


7 41 


7-80 


8-16 


849 


8J 


4-62 


5-28 


5-82 


6-27 


6-63 


6-99 


7-32 


7-89 


837 


8-82 


9-29 


9 59 


9 


5-16 


5-91 


6-51 


7-02 


7-47 


7-S6 


8-22 


8-85 


9-39 


9-90 


10-35 


1077 


^ 


5-76 


6-60 


7-26 


7-SO 


8-37 


8-76 


9*15 


9-84 


10-47 


11-01 


11-52 


11-98 


10 


6-89 


7-32 


8-04 


8-67 


9*21 


9-69 


10-14 


10-92 


11*61 


12*21 


12-78 


13-29 


10} 


7-05 


8-04 


8-S8 


9*54 


10*14 


10-68 


11*16 


12-03 


12-78 


1347 


14-07 


14-64 


11 


7-74 


8-85 


9*72 


10-47 


11-31 


11-73 


12*45 


13-20 


14-04 


14-76 


15-45 


1605 


•U* 


8-43 


9-66 


10*B2 


11-46 


12*15 


12*78 


13*80 


14-61 


15-33 


16-14 


16-88 


17-56 


12 


9-18 


10-53 


11*58 


12-48 


13-26 


13*95 


14-58 


15*72 


16*71 


17*58 


18-39 


19-111 


12} 


9-96 


11-40 


12-57 


13-53 


14-37 


15*15 


15-84 


17*04 


1812 


19-08 


19-92 


20-73 


1? 


10-S0 


12-36 


13-59 


14-64 


15-57 


16-38 


16-92 


18*45 


19-59 


20-64 


21-57 


22-44 


is* 


11-64 


13-32 


14-64 


15*78 


16*77 


17-67 


18-48 


19*89 


21*15 


22-26 


23 25 


2419 


14 


12-51 


14-31 


15-75 


16*98 


18*03 


18-99 


19-86 


21*39 


22-74 


23*94 


25-02 


26-01 


14} 


13-41 


15-56 


16-92 


18-21 


19*35 


20-37 


21-30 


22-95 


24-39 


25-62 


26-83 


27-90 


15 


14-31 


16-44 


18-09 


19-50 


20*70 


21-81 


22-80 


24-57 


26-10 


2748 


28-71 


29 *88 


16 


16-35 


18-69 


20-58 


22-17 


23*58 


24-81 


25-95 


27-93 


29-70 


31-26 


32-67 


33-99 


17 


1845 


21-12 


23*25 


25-05 


26-58 


2S-02 


29-28 


31-56 


33-57 


35-28 


86-90 


38-37 


IS 


20-67 


23-67 


26*04 


28-08 


29*82 


31*41 


32-82 


35-37 


37-59 


39-57 


41-37 


43-02 


19 


23-04 


26-37 


29*04 


31-26 


33-51 


34-98 


36-57 


39-39 


41-88 


44-07 


46-08 


47-94 


20 


25-53 


29-22 


32-16 


34-65 


36-81 


38-76 


40-53 


43-65 


46-38 


4S-84 


51-06 


53-10 


22 


30-90 


35-37 


88-91 


41-94 


44-55 


46-S9 


49-86 


52-95 


56-13 


59-10 


61-80 


64-26 


24 


36-78 


42-09 


46-32 


49-89 


53-01 


55-S3 


58-35 


62-S5 


66-81 


70-32 


73-53 


76-47 


20 


43-17 


49-38 


54-36 


5S-56 


62-25 


65-52 


67-68 


73*80 


78 42 


82-53 


86-34 


89-76 


28 


50-04 


57-27 


63-06 


67-92 


72-18 


75-99 


79-44 


85-56 


90-93 


95-70 


100-1 


1040 


30 


57*45 


65-76 


72-39 


77-97 


82-86 


87-21 


91-20 


98-22 


104-4 


109-9 


114-9 


119 5 


82 


65-37 


74-88 


82-53 


88-71 


94-26 


99-24 


103-7 


111-8 


118-7 


125-0 


130-7 


1360 


84 


73-80 


84*48 


92-97 


100-2 


106-3 


112-0 


117-1 


126-2 


134-0 


141-1 


147-6 


153-5 


86 


82-71 


94-6S 


104-2 


1122 


119-3 


125-6 


131-3 


141-4 


150-3 


158-2 


165-4 


172-1 


38 


92-16 


105-5 


116-1 


125-0 


134-0 


136-9 


146-3 


157-6 


167-5 


176-3 


184-3 


191-7 


40 


102-1 


116-9 


129-6 


13S-6 


147-3 


155-1 


162-1 


174-6 


185-6 


195-3 


204-2 


212 4 


42 


112-6 


128-9 


141-8 


152-8 


162*4 


170-9 


178-7 


192-5 


204-6 


215-3 


225-2 


234-2 


44 


123-5 


141*4 


155-7 


167*7 


178*1 


1S7*6 


199-4 


211-3 


224*5 


2363 


247-1 


257-0 


46 


135-0 


154*6 


170-1 


183-3 


194-6 


204-6 


214-3 


230-0 


2454 


258-3 


2701 


280 9 


48 


147-0 


168-3 


185-3 


199-6 


212-1 


223 2 


233-4 


251-5 


267-2 


281-3 


294-1 


306-0 


50 


159-6 


182*6 


201-0 


216-5 


230-1 


242-3 


253-3 


272-9 


289-9 


305-1 


319-2 [331-8 


52 


172-6 


197-6 


217-4 


234-2 


249-0 


262-0 


270-7 


295-2 


313-5 


330-0 


345-3 ;358-8 


54 


186-1 


213-0 


234-5 


252-6 


268-4 


282-6 


295-4 


318-3 


338-1 


356-1 


372-3 1387-0 


56 


2C01 


229-1 


252-2 


271-6 


2S8-7 


303-9 


317-7 


342-3 


363-6 


382-8 


400*2 416-4 


; ^s 


214-7 


245*8 


270-5 


291-4 


309-6 


325-8 


340-8 


367-2 


3S9-7 


410-1 


429-3 


446-1 


I GO 


229-8 


263-0 


2S9-5 


311-7 


331-2 


348-9 


3G4-8 


393-0 


417-3 


439-5 


459 6 


477-9 


i 70 


312-8 


357-9 


393-9 


424-5 


451-2 


474-9 


496-5 


5346 


568 2 


598-2 


625-5 


6504 



210 



PEOPOETI02>3 OF STEA3J-EX&IXES. 



X01TD7AL HOESE POWEE OF LOTT PRESSURE ENGINES. 



P 3 - •! 
III 








LENGTH O 


P stuoks rs~ teei 










1 


H 


2 


H 


3 

•49 


•52 


4 


5 


6 


7 


8 


9 


4 


■35 


•39 


•43 


•46 


•54 


•53 


•62 


•65 


•68 


•70 


5 


•53 


•61 


•67 


•72 


•76 


•81 


.64 


•91 


■96 


1-02 


1-06 


111 


6 


'76 


■ST 


•96 


1-04 


1-10 


1-16 


1-22 


1-81 


1-39 


1-47 


1-53 


1-59 


T 


1-04 


1-19 


1-31 


1-41 


1-50 


1-58 


1-65 


1-78 


1-59 


1-99 


2-09 


2-17 


S 


1-36 


1-56 


1-72 


1-85 


1-96 


2.07 


2-16 


2-33 


2-47 


2-60 


2-72 


•2 -S3 


9 


1-72 


1-97 


2-17 


2-34 


249 


2-62 


2-74 


2-95 


3-13 


3-30 


3-45 


8-59 


10 


2-13 


2-44 


2-63 


2-S9 


3-07 


3-23 


3-33 


3-64 


3-S7 


4-07 


4-26 


4-43 


11 


2-57 


2-95 


3-24 


3-49 


8'77 


3-91 


4-15 


440 


4-68 


4-92 


5-15 


5-35 


12 


3-06 


3-51 


3-86 


4-16 


442 


4-65 


4-86 


5"24 


5-57 


5-56 


6-13 


6-37 


13 


3-60 


4-12 


4-53 


4-88 


519 


5-46 


5-64 


615 


6-53 


6-83 


7-19 


7-48 


14 


417 


4-77 


5-25 


5-66 


601 


6-33 


6-62 


7-13 


7-58 


7-SS 


8-34 


8-67 


15 


4-77 


5-43 


6-03 


650 


6-90 


7-27 


7-60 


8-19 


8-70 


9-16 


9-57 


Q-96 


16 


5-45 


6-23 


6-36 


7-39 


7-S6 


S-27 


8-65 


931 


9-90 


10-42 


10-89 


11-33 


17 


615 


7-04 


7-75 


8-35 


8-56 


9 34 


9-76 


10-52 


11-17 


11-76 


12-30 


12-79 


18 


6-S9 


7-89 


8-63 


9-36 


9 94 


1047 


10-94 


11-79 


12-53 


13-19 


13-79 


14-34 


19 


7-68 


6-79 


9-68 


10.42 


11-17 


11-66 


12-19 


13-13 


13-96 


14-69 


15-86 


15-93 


20 


8-51 


9-74 


10-72 


11-55 


12-27 


12-92 


13-51 


14-55 


1546 


16-23 


17-02 


17-70 


22 


10-30 


11-79 


12-97 


13-98 


14-85 


15-63 


16-62 


17-65 


18-71 


19-70 


20-60 


2142 


24 


12-26 


14-03 


15-44 


16-63 


17-67 


13-61 


19-45 


20-95 


22-27 


23-44 


24-51 


25-49 


26 


14-39 


16-46 


1812 


19-52 


20-75 


21-84 


22-56 


24-60 


26-14 


27-51 


2S-73 


29-92 


23 


16-63 


19-09 


21-02 


22-64 


24-06 


25-33 


2643 


23-52 


30-31 


31-90 


33-36 


34-69J 


80 


19-15 


21-92 


24-18 


25-99 


27-62 


29-07 


8O40 


82-74 


84-80 


36-63 


88-30 


39-83 


32 


21-79 


24-96 


27-51 


29-57 


3142 


33-03 


84-59 


37-26 


39-59 


41-63 


43-57 


45-32! 


34 


24-60 


28-16 


30-99 


33-39 


35-44 


37-34 


39-04 


42-06 


44-69 


47-05 


49-19 


5116J 


36 


27-57 


31-50 


34-74 


37-42 


39-77 


41-57 


43-77 


47-15 


50-11 


52-75 


55-15 


57-361 


38 


30-72 


85-17 


£5-71 


41-69 


44-66 


46-64 


4S-77 


52-54 


55-83 


58-73 


6145 


63-91 i 


40 


34-04 


3S-97 


42-89 


46-20 


49-10 


51-69 


54-04 


58-21 


61-86 


65-12 


63-08 


70-81 


42 


37-53 


42-96 


47-29 


50-94 


54-13 


56-98 


59-53 


64-18 


68-21 


71-78 


75-06 


7S-06! 


44 


41-19 


47-15 


51-90 


55-91 


50-35 


62-54 


6G46 


70-44 


74-85 


75-79 


S2-3S 


S5-CS! 


46 


45-02 


51-54 


56-72 


61-10 


64-83 


63-19 


71-43 


76-69 


SI -81 


86-12 


90-04 


g i-u 


43 


49-02 


58-11 


61-76 


6653 


70-70 


74-42 


77-52 


83-83 


89-08 


93-75 


95-04 102-0 1 


50 


53-19 


60-39 


67-02 


7219 


70-71 


80-76 


84-44 


90-96 


96-65 


101-7 


1064 


110-6 ! 


52 


57 ^55 


65-86 


72-43 


18-08 


S3 -00 


87-85 


90-25 


9S-40 


104-5 


110-0 


115-1 


119-6 


54 


62-04 


71-02 


7S-17 


34-20 


6943 


94-20 


9349 


106-1 


11-2-7 


113-7 


124-1 


129 


56 


66-72 


76-3S 


84-07 


90-55 


96-23 


101-30 


105-9 


1141 


121-2 


127-6 


1334 


138-8 


5S 


71-53 


81-93 


9018 


97-14 


103.2 


105-6 


113-6 


122-4 


129-9 


136-7 


143-1 


14-^-7 


60 


76-60 


87-68 


96.50 


103-9 


1104 


116-3 


121-6 


131-0 


139-2 


146-5 


153-2 


159 -3 


62 


81-79 


93-62 


103-04 


111-0 


117.9 


124-13 


129-81 


139-3 


148-6 


156-7 


163-6 


170-3 


64 


87-15 


99-S4 


no-o 


113-3 


125-7 


132-3 


13S-3 


149-0 


1554 


166-7 


174-3 


151-3 


66 


92-63 


106-1 


116-8 


125-3 


133-6 


140-7 


147-3 


155-5 


168-4 


177'8 


1554 


192-8 


63 


93-40 


112-6 


123-9 


133-6 


141-3 


149-4 


156 2 


168-2 


173-8 


18S-2 


196-S 


204-6 


70 


104-26 


119-3 


131-3 


141-5 


150-4 


158-3 


165-5 


175-2 


169-4 


199-4 


203-5 


216-8 


72 


110-30 


126-2 


139-0 


149-7 


159-1 


167-4 


175-1 


1SS-6 


200-4 


211-0 


220-6 


229-4 | 


74 


116-5 


133-4 


146-8 


158-1 


167-9 


176-7 


165-4 


199-2 


211-6 


223-4 


233-0 


242*2 ' 


76 


122-9 


140-7 


154-8 


166-8 


173-6 


136-6 


1950 


210-1 


223-3 


235-1 


245-3 


255-6 i 


73 


1294 


143-2 


163-1 


175-6 


1S6-7 


196-5 


205-4 


2-21-4 


235-2 247-6 


255-9 


269"2 ' 


80 


136-2 


155-8 


171-6 


134-3 


196-4 


206-7 


216-1 


232-S 


2474 260-5 


272-3 


253-2 


62 


143-0 


163-S 


180-2 


194-2 206-2 


217-3 


226-9 


244-6 


260-0 273-3 


2861 


297* 


34 


150-1 


171-3 


139-1 


203-8 


216-5 


227-9 


235-3 


256-7 


272-8 


2 =7-1 


800 2 


312-2 j 


1 86 


157-4 


150-1 


19S-2 


213-6 


227-0 237-5 


247-4 


269-1 


236-0 


801-0 


314-7 


827-3 I 


! 88 


164-3 


133-6 


207-6 


223-6 


2375 


250-2 


261-6 


231-7 


•299-4 


315-2 


329-5 


842-7 


90 


172-3 


197-3 


217-1 


233-9 


243-6 


261-7 


2786 


294-7 


313-2 829-7 


844-7 


85S'5 


100 


, 212-8 


243-5 


265-0 


288-8 306.3 


323-0 337-7 863-8 3S6-6 407 "0 


425-5 


442-6 



RULES FOR FINDING THE HORSES POWER. 211 

Example 2. — What is the nominal power of a low pressure 
engine of 40 inches diameter of cylinder and 5-feet stroke. 

Here 40 x 40 = 1,600, which multiplied by T71 — which is 
the cube root of 5 very nearly — we get 2,736, which divided by 
47 gives 58'21 as the nominal horse power. 

Tbe actual horse power of an engine is determinable by the 
application of an instrument to determine the amount of power 
it actually exerts. The mode of determining this will be ex- 
plained hereafter. Meanwhile it may be repeated that an actual 
horse power is a dynamical unit capable of raising a load of 
33,000 lbs. one foot high in each minute of time. The nominal 
power of a high pressure engine may be taken at three times 
that of a low pressure engine of the same size. 

The assumed pressure in computing the nominal power of 
low pressure engines is 7 lbs. on each square inch of the piston, 
and the assumed pressure in computing the nominal power of 
high pressure engines is 21 lbs. on each square inch of the piston. 
The assumed speed of the piston varies with the length of stroke 
from 160 to 256 feet per minute, namely, for a 2 ft. stroke, 
160 ft. ; 2ift., 170 ; 3 ft., 180 ; 4 ft., 200 ; 5 ft., 215 ; 6 ft., 228; 
7 ft., 245 ; and 8 ft., 256 feet per minute. 

In point of fact, in all modern low pressure engines the un- 
balanced pressure of steam upon the piston is much more than 
7 lbs., and in most modern high pressure engines the unbalanced 
pressure of steam upon the piston is much more than 21 lbs. The 
speed of the piston is also frequently much more than 256 feet 
per minute. In the case of screw engines the Admiralty employs 
a rule to determine the power, in which the old assumed pressure 
of 7 lbs. per square inch is retained, but in which the actual speed 
of piston is taken into account. This rule is as follows : — 

ADMIRALTY RULE FOE DETERMINING THE NOMINAL POWER OF AN 

ENGINE. 

Rule. — Multiply the square of the diameter of the cylinder in 
inches oy the speed of the piston in feet per minute^ and 
divide oy 6.000. The quotient is the nominal power. 

Example. — What is the power of an engine with a cylinder 



212 TEOFOETIOXS OF STEAM-ENGIS" Z ; . 

of 42 inches diameter, and 3-| feet stroke, and which makes 85 
revolutions per minute ? 

Here 42 x 42 = 1,764. The length of a double stroke will 
be o-l x 2 = 7 feet, and as there are 85 reTolntions or doable 
strokes per minute, 85 x 7 = 595 will he the speed, of the j:s: : e 
in feet per minnte. Xow 1,764 x 595 = 1,049,580, which, di- 
vided by 6,000 = 175 horses power. 

The area of the piston in circular inches, it will be recollect- 
ed, is found by multiplying the diameter by itself. Thus a pis- 
ton 50 inches diameter contains 50x50, or 2, 500 circnlar indi£& 
IS ow as every circnlar inch is '7854 of a square inch, we must, 
in order to find the area of the piston in square inches, multiply 
the diameter by itself and by • 7 B : -. ~ hich will give the area in 
square inches. Thus, 2,500 x *7854 = 1,963*5 sqnare inches, 
which is tlie area in square inches of a piston 50 inches in diam- 
etei , The circumference of any circle is obtained, by multiply- 
ing the diameter by 3T416. Hence the length of a string oi 
tape that will be required to encircle a piston 50 inches in diam- 
eter will be 50 x S'1416 = 157*08 inches. The areas of pumps, 
pipes, safety-valves, and all other circular objects, is computed in 
the same way as the areas of circles or pistons. Some valves 
are annular valves, consisting not of a flat circular plate, but of 
a ring or annnlus of a certain breadth. To compute the area 
of such a valve we must first compute the area of the outer 
circle, and then the area of the inner, and subtract the less from 
the greater, which will give the area of the annnlus. So in like 
manner, in trunk engines, we must subtract the area of :"_r 
trunk from the area of the piston. 

GENERAL CONSIDERATIONS AND INSTRUCTION. 

In proceeding to design an engine for any given purpose, 
the nominal power may either be fixed or the nominal power 
may be left indeterminate, and only the work be fixed vr ;_■;;. 
the engine has to perform. In the first case we have only to 
ascertain by the foregoing rules or tables what the dimensions 
of a cylinder are which correspond to the nominal p 



DRAWINGS SUITABLE FOR ALL POWERS. 213 

and we Lave then to make all the other parts of dimensions 
corresponding thereto, which we shall be enabled to do by the 
rules here laid down. Of course the engineer settles for himself 
some particular type of engine which he prefers to adopt as the 
one that is to govern his practice, and any drawing of an engine 
of a given size or power is applicable to the construction of a 
similar engine of any other size or power by merely altering the 
scale of the drawing. If, therefore, any engineer decides upon 
the class of land engine, paddle engine, or screw engine which 
he prefers to construct, and chooses to get a set of drawings of 
such engine on any given scale lithographed, such drawirgs will 
be applicable to all sizes and powers of that class of engine by 
altering the scale in the proportion rendered necessary by the 
enlarged or diminished diameter of the cylinder answerable to 
the required power. Thus, if we have a drawing of a marine 
engine of 32 inches diameter of cylinder and 4-feet stroke, made 
to the scale of -|-inch to the foot, we may from such drawing 
construct a similar engine of 64 inches diameter and 8-feet 
stroke by merely altering the scale to one of J-inch to the foot, 
so that every part will in fact measure twice what it measured 
before. In order to make the same drawing applicable to any 
size of engine, whether large or small, we have only to divide 
the diameter of the cylinder into the number of parts that the 
cylinder is to have of inches, and then we may use the scale so 
formed for the scale of the drawing. Thus, if we wish the 
engine to have a cylinder of 30 inches diameter, we must divide 
the diameter of the cylinder as shown in the drawing into 30 
equal parts, each of which will represent an inch, and of course any 
twelve of them will represent a foot. If we now measure anj 
other part of the engine, such as the diameter of the air pump, 
diameter of crank shaft, or any other part by this scale, we shaL 
find the proper dimensions of the part in question, If we wish 
to construct from the drawing an engine of 60 or 100 inches, 
and of corresponding stroke, we have only to divide the diam- 
eter of the cylinder into 60 or 100 equal parts, and use each of 
those parts as an inch of the scale, when the proper dimensions 
of all the \m\o will be at once obtained. 



814 PSOFOBiioys •:: siEAX-nxGrsxs. 

It will be needless to guard these remarks against the obvious 
-z:r7 r i:i. :b.it Lr. :.sb :: ~b7~ :.-b ar_i ~b~ ~zi:~l engine? i~. 
will be proper to make such slight modifications in some of the 
details as will conduce to greater convenience in working or in 
construction- For instance, as the height and strength of a man 
are a given quantity, it will obviously not be proper in doubling 
the size of all the other parts to double the height : the starting 
handles, or even to double their strength. In the case of oscil- 
lating engines, again, with a crank in the intermediate shar:. ft 
may be difficult to get a sound crank made in the case of very 
large engines, and some other expedient may have to be adopted. 
Again, in the case of very nail engines, the flanges and bolts 
require to be a little larger than the proportion derived 
from a drawing of large engines, and the valve chesr; :: the 
teed pumps and other parts may be too small if made strictly to 
scale to get the band into earavjemomSjr to clear them out. All 
such points however are matters of practical convenience, only 
to be determined by the thoughtfulness and experience of the 
engineer, and in nowise affect the main conclusion that a draw- 
ing of an engine of any one size will suffice for the construction 
of engines of other sizes by merely changing the scale. It wiP 
consequently save much trouble in drawing offices to have one 
certain type of engine of each kind lithographed in all its details, 
and then engines of aD sizes may be made therefiom by adding 
the proper scale, and hy marking upon the drawing the proper 
dimensions of each part in feet or inches — the measurements 
being taken from a table fixed once for aD, either by computation 
or by careful measurement of the drawing with the different 
suitable scales. B~ thus system ::ising the work of the drawing 
:zi:e. Ia_-: _ .r —:~ 1 e = ;-t1 ar_'l z::::a:- -r.-r~b7r.r-l. 

I: easy to understand the principle on which the main parts 
■: :.zb b7--'.7-b in" s: lb ~_::7 :rrl :r_e-L ~b zlzlsz in :"_e ±-: ~~.: - 
have the requisite quantity of boiler surface :: generate the 
steam, the requisite quantity of water sent into the boil-. - 
::;: up the proper supply, and the requisite quantity of cold 
Wubb7 :o Cr-lbusb :le =:eiii afrei :: 1. - -i~e_ r^;ri:z :: .-b 
piston. In common boilers about 10 square feet of heating 



GENERAL CONSIDER ATIONS AND INSTRUCTIONS. 215 

surface will boil off a cubic foot of water in the hour, and this 
in the older class of engines was considered the equivalent of a 
horse power. At the atmospheric pressure, or with no load on 
the safety valve, a cubic inch of water makes about a cubic foot 
of steam ; and at twice the atmospheric pressure, or with 15 lbs. 
per square inch on the safety valve, a cubic inch of water will 
make about half a cubic foot of steam. For every half cubic 
foot of such steam therefore abstracted from the boiler there 
must be a cubic inch of water forced into it. So if we take the 
latent heat of steam in round numbers at 1,000 degrees, and if 
the condensing water enters at 60°, and escapes at 100°, the 
condensing water has obviously received 40 degrees of heat, and 
it has received this from the steam having 1,000° of heat, and 
the 112° which the steam if condensed into boiling water would 
exceed the waste- water in temperature. It follows that in order 
to reduce the heat of the steam to 100° there must be 1,112° of 
heat extracted, and if the condensing water was t only to be 
heated 1 degree, there would require to be 1,112 times the 
quantity of condensing water that there is water in the steam. 
Since, however, the water is to be heated 40°, there will only 
require to be one-fortieth of this, or about -g^th the quantity of 
injection water that there is water in the steam. These rough 
determinations will enable the principle to be understood on 
which such proportions are determined. The proportions of the 
condenser and of the air-pump were determined by Mr. Watt 
at one-eighth of the capacity of the cylinder. In more modern 
engines, and especially in marine engines where there are irregu- 
larities of motion, the air-pump is generally made a little larger 
than this proportion, and with advantage. The condenser is 
also generally made larger, and many engineers appear to con- 
sider that the larger the condenser is the better. Mr. Watt, 
however, found that when the condenser was made larger than 
one-eighth of the capacity of the engine the efficiency of tlio 
engine was diminished. The fly-wheel employed in land engines 
to control the irregularities of motion that would otherwise 
exist, is constructed on the principle that there shall be a revolv- 
ing mass of such weight, and moving with such a velocity, as to 



216 PROPORTIONS OP STEAM-ENGINES. 

constitute an adequate reservoir of power to redress irregulari- 
ties. It is found that in those cases where the most equahle 
motion is required, it is proper to have as much power treasured 
up in the fly- wheel as is generated in 6 half-strokes, though in 
many cases the proportion is not more than half this. It is quite 
easy to tell what the weight and velocity of the fly-wheel must 
he to possess this power. "When we know the area of the piston 
and the unbalanced pressure per sq. inch, we easily find the 
pressure urging it, and this pressure multiplied by the length of 
6 half-strokes represents the amount of power which, in the 
most equahle engines, the fly-wheel must possess. Thus, suppose 
that the pressure on the piston were a ton, and that the length 
of the cylinder were 5 feet, then in 6 half-strokes the space 
described by the piston would be 30 feet. The measure of the 
power therefore is 1 ton descending through 30 feet, and if there 
were any circumstance which limited the weight of the fly- 
wheel to 1 tpn, then the velocity of the rim — or more correctly 
of the centre of gyration — must be equal to that which any 
heavy body would have at the end of the descent by falling from 
a height of 30 feet, and which velocity may easily be determined 
by the rule already given for ascertaining the velocity of falling 
bodies. If the weight of the fly-wheel can be 2 tons, then the 
velocity of the rim need only be equal to that of a body falling 
through 15 feet, and so in all other proportions, so that the 
weight and velocity can easily be so adjusted as to represent 
most conveniently the prescribed store of power. 

With these preliminary remarks it will now be proper to 
proceed to recapitulate the rules for proportioning all the parts 
of steam engines illustrated by examples : — 

STEAM PORTS. 

The area of steam port commonly given in the best engines 
working at a moderate speed is about 1 square inch per nominal 
horse-power, or ^th of the area of the cylinder, and the area of 
the steam pipe leading into the cylinder is less than this, or *66 
square inch per nominal horse power. Since however engines 



PROPER AREAS OP CYLINDER PORTS. 217 

are now worked at various rates of speed it will be proper to 
adopt a rule in whicli the speed of the piston is made an element 
of the computation. This is done in the rules which follow hoth 
for the steam port and branch steam pipe. 

TO FIND THE PROPER AEEA OF THE STEAM OE EDUCTION FORI 
OF THE CYLINDER. 

fipLE. — Multiply the square of the diameter of tlie cylinder in 
inches dy the speed of the piston in feet per minute and ~by 
the decimal *032, and divide the product dy 140. The quo- 
tient is the proper area of the cylinder port in square inches. 

Example.— yfliak is the proper area of each cylinder port in 
an engine with 64-inch cylinder, and with the piston travelling 
220 feet per minute ? 

Here 64 x 64 = 4,096, which multiplied by 220 = 901,120, 
and this multiplied by *032 = 28,835*8, which divided by 140, gives 
206 inches as the area of each cylinder port in square inches. 

This is a somewhat larger proportion than is given in some 
excellent engines in practice. But inasmuch as the application 
of lap to the valve virtually contracts the area of the cylinder 
ports, and as the application of such lap is now a common prac- 
tice, it is desirable that the area of the ports should be on the 
large side. In the engines of the 'Clyde,' 'Tweed,' ' Tay,' and 
'Teviot,' by Messrs. Oaird and. Co., the diameter of the cylinder 
was 74| inches, and the length of the stroke 7-§- feet, so that the 
nominal power of each engine was about 234 horses. The cyl- 
inder ports were 33^ inches long and 6| inches broad, so that 
the area of each port was 224*4 square inches, being somewhat 
less than the proportion of 1 square inch per nominal horse 
power, but somewhat more than the proportion of ^ T th of the 
area of the cylinder. As the areas of circles are in the propor- 
tion of the square roots of their respective diameters, the area 
ot a circle of one-fifth of the diameter of the piston will have 
one-twenty-fifth of the area of the piston. One-fifth of 74£ths 
is 15 nearly, and the area of a circle 15 inches in diameter is 
176*7 square inches, which \% considerably less than the actual 
10 



218 PROPORTIONS OF STEAM-ENGINES. 

area of tlie port. By the rule we have given the area of the 
ports of this engine would, at a speed of 220 feet per minute, be 
about 277 square inches, which is somewhat greater than the 
actual dimensions. At a speed of the piston of 440 feet per 
minute the area of the port would be double the foregoing. 

STEAM PIPE. 

In the engines already referred to, the internal diameter of 
each steam pipe leading to the cylinder is 13f inches, which 
gives an area of 145*8 square inches. It is not desirable to make 
the steam pipe larger than is absolutely necessary, as an increased 
external surface causes increased loss of heat from radiation. 
The following rule will give the proper area of the steam pipe 
for all speeds of piston : — 

TO FIND THE AEEA OF THE STEAM PIPE LEADING TO EACH 

CTLINDEE. 

Rule. — Multiply the square of the diameter of the cylinder in 
inches oy the speed of the piston in feet per minute and oy the 
decimal '02, and divide the product oy 170. The quotient is 
the proper area of the steam pipe leading to the cylinder in 
inches. 

Example. — "What is the proper area of the branch steam pipe 
leading to each cylinder in an engine with a cylinder 74J inches 
diameter, and with the piston moving at a speed of 220 feet per 
minute ? 

Here 74*5 x 74*5 = 5,550*25, which multiplied by 220 == 
1,221,055. and this multiplied by *02 = 24,421*1, which divided 
by 170 = 144 square inches nearly. The diameter of a circle of 
144 square inches area is a little over 13-J inches, so that 13£ 
inches would be the proper internal diameter of each branch 
steam pipe in such an engine. The main steam pipe em- 
ployed in steamers usually transmits the steam for both the 
engines to the end of the engine-house, where it divides into 
two blanches — one extending to each cylinder. The main steam 
pipe will require to have nearly, but not quite, double the 
area of each of the branch steam pipes. It would require to 



PROPER AREA OF SAFETY VALVES. 219 

have exactly double the area, only that the friction in a large 
pipe is relatively less than in a small; and as, moreover, the 
engines work at right angles, so that one piston is at the end of 
its stroke when the other is at the beginning, and therefore 
moving slowly, it will follow that when one engine is making 
the greatest demand for steam the other is making very little, 
so that the area of the main steam pipe will not require to be as 
large as if the two engines were making their greatest demand 
at the same time. 

SAFETY YALVES. 

It is easy to determine what the size of an orifice should be 
in a boiler to allow any volume of steam to escape through it in 
a given time. For if we take the pressure of the atmosphere at 
15 lbs., and if the pressure of the steam in the boiler be 10 lbs. 
more than this, then the velocity with which the steam will flow 
out will be equal to that which a heavy body would acquire in 
falling from the top of a column of the denser fluid that is high 
enough to produce the greater pressure to the top of a column 
of the same fluid high enough to produce the less pressure, and 
this velocity can easily be ascertained by a reference to the law 
of falling bodies. In practice, however, the area of safety valves 
is made larger than what answers to this theoretical deduction, 
partly in consequence of the liability of the valves to stick round 
the rim, and because the rim or circumference becomes relatively 
less in the case of large valves. One approximate rule for safety 
valves is to allow one square inch of area for each inch in the di- 
ameter of the cylinder, so that an engine with a 64-inch cylinder 
would require a safety valve on the boiler of 64 square inches 
area, which answers to a diameter of about 9 inches. The rule 
should also have reference, however, to the velocity of the piston, 
and this condition is observed in the following rule: — 

TO FEND THE PEOPEE DIAMETEE OF A SAFETY VALVE THAT WILL 
LET OFF ALL THE STEAM FEOM A LOW PEESSUEE BOILEE. 

Rule.— Multiply the square of the diameter of the cylinder in 
inches oy the speed of the piston in feet per minute, and 



220 PROPORTIONS OF STEAM-ENGINES. 

divide the 'product by 14,000. The quotient is the proper 
area of the safety valve in square inches. 

Example. — "What is the proper diameter of the safety valve 
of a boiler that supplies an engine "with steam, having a 64-inch 
cylinder, and with the piston travelling 220 feet per minnte? 

Here 04 x 64 = 4,096, which multiplied by 220 = 901,120, 
and this divided by 14,000 = 64*3, which is the proper area of 
the safety valve in square inches. 

AXOTHEE EULE FOE SAFETY VALVES. 

Multiply the nominal horse power of the engine by "375, and to 
the product add 16*875. The sum is the proper area of ilie 
safety naive in square inches, when the boiler is low pressure. 

Example. — What is the proper diameter of the safety valve 
for a low pressure engine the nominal power of which is 140 
horses ? 

Here 140 x '375 = 52-5, adding to which the constant num- 
ber 16*875, we get 69-375, which is the proper area of the safety 
valve in square inches for a low pressure engine. 

A 60-inch cylinder and 6-feet stroke is equal to 140 nominal 
horses power, so that this rule gives somewhat more than a 
square inch of area in the valve for each inch of diameter in the 
cylinder in that particular size of engine. 

The opening through the safety valve must be understood to 
be the effective opening clear of bridges or other obstacles, and 
the area to be computed is the area of the smallest diameter of 
the valve. Most safety valves are made with a chamfered edge, 
which edge constitutes the steam tight surface, and the effective 
area is what corresponds to the smaller diameter of the valve 
and not to the larger. All boilers should have an extra or ad- 
ditional safety valve of the same capacity as the other, which 
may act in case of accident to ihe first from getting jammed or 
otherwise. The dimensions of safety valve here computed is 
that adequate for letting off all the steam. But in some cases 
the whole steam is not supplied from one boiler, and a safety 
f alve in such case must be put on each boiler, but of a less area, 



PROPER DIAMETER OF THE FEED PIPE. 221 

in proportion to the smaller volume of steam it has to let off. 
If there are two boilers, the safety valve on each will be half 
the area of the foregoing ; if three boilers, one-third of the area ; 
if four boilers, one-fourth of the area ; and so of all other pro- 
portions. The area of the waste steam pipe should be the same 
as that of the safety valve. 

TO FIXD THE PROPER DIAMETER OF THE FEED PIPE. 

Rule. — Ifultijily the nominal horse power of the engine as com-' 
puted by the Admiralty rule ly '04, to the product addS] 
extract the square root of the sum. Th-e result is the diam- 
eter of the feed pipe in inches. 

Example 1. — What is the proper diameter of the feed pipe in 
inches of an engine whose nominal horse power is 140 ? 

140 = nominal horse power of engine 
•04 = constant multiplier 



5-6 

3 = constant to be added 



8-6 



and ^8*6 = 2*93 diameter of feed pipe in inches. 



Example 2. — What is the proper diameter of the feed pipe 
m inches in the case of an engine whose nominal horse power 

is 385 ? 

385 = nominal horse power of engine 
•04 == constant multiplier 



15-4 

3 constant to be added 



18'4 



and yi8*4 = 4-29 diameter of feed pipe in inches. 



222 PROPORTIONS OF STEAM-EXGTSTS. 

TO FIND THE PROPER DIMENSIONS OF THE AIR PUMP AND 

CONDENSER. 

In land engines the diameter of the air pump is made half 
that of the cylinder, and the length of stroke half that of the 
cylinder, so that the capacity is -Jth that of the cylinder ; and 
the condenser is made of the same capacity. Bnt in marine en- 
gines the diameter of the air pnmp is made # 6 of the diameter 
of the cylinder, and the length of the stroke is made from "57 to 
•G times the stroke of the cylinder, and the condenser is made 
at least as large. In some cases the air pnmp is now made dou- 
ble-acting, in which case its capacity need only be half as great 
as when made single-acting. 

TO FIND TEE PROPER AREA OF THE INJECTION PIPE. 

Rule. — Multiply the nominal horsepower of the engine, as com- 
puted oy the Admiralty rule, dy 0*69, and to the product 
add 2*81. The sum is the proper area of the injection pipe 
in square inches. 

Example 1. — "What is the proper area of the injection pipe in 
square inches of an engine whose nominal horse power is 1-40 ? 

140 = nominal horse power of engine 
•069 = constant multiplier 



9-66 

2-81 = constant to be added 



Answer 12*4? = area of injection pipe in square inches. 



Example 2. — What is the proper area of the injection pipe in 
square inches of an engine whose nominal horse power is 385 ? 

3S5 = nominal horse power of engine 
•069 = constant multiplier 



26-56 

2 - Sl = constant to be added 



Answer 29-37 = area of injection pipe in square inches. 



PROPER AREA OF THE FOOT VALVE PASSAGE. 223 

The area of the injection orifice is usually made about l-250th 
part of the area of the piston, which, in an engine of 385 horse 
power, would be about 27*7 inches of area. For warm climates 
the area should be increased. 

TO FIND T1IE PKOPER AREA OP THE FOOT VALVE PASSAGE. 

Rule. —Multiply the nominal horse poicer of the engine by 9, 
divide the product ~by 5, add 8 to the quotient. The sum w 
the proper area of foot valve passage in square inches. 

Example 1. — What is the proper area of the foot valve pas- 
sage in square inches of an engine whose nominal horse power 
is 140? 

140 = nominal horse power of engine 
9 = constant multiplier 



constant divisor 5)1260 



252 

8 = constant to be added 



Answer 260 = area of foot valve passage in square inches. 



Example 2. — What is the area of foot valve passage in square 
jiches of an engine whose nominal horse power is 385 ? 

385 =r nominal horse power of engine 
9 = constant multiplier 



constant divisor 5)3465 



693 
8 = constant to be added 



Answer 701 = area of foot valve passage in square inches. 



The discharge valve passage is made of the same size as the* 
loot valve passage. 

A common rule for the area of the foot and discharge valve 
passages is one-fourth of the area of the air pump, and the waste 



224 PROPORTIONS OP STEAM-ENGINES. 

water pips is made one-fourth of the diameter of the cylinder, 
which gives a somewhat less area than that through the foot and 
discharge valve passages. Such rules, however, are only appli- 
cable to slow-going engines. In rapid-working engines, such aa 
those employed for driving the screw propeller by direct action, 
and in which the air-pump is usually double acting, the area 
through the foot and discharge valves should be equal to the 
area of the air-pump, and the waste water pipe should also have 
the same area. In all cases, therefore, in which these or other 
rules dependent on the nominal power are applied to fast-going 
engines, the nominal power must be computed by the Admiralty 
rule, in which the speed of the piston is taken into account. 

TO FIND THE PROPER DIAMETER OF THE WASTE WATER 

PIPE. 

Rule. — Multiply the square root of the nominal horse power of 
the engine oy 1*2. The product is the diameter of the waste 
water pipe in inches. 

Example 1. — What is the diameter of the waste water pipe, 
in inches, of an engine whose nominal horse power is 140 ? 

140 = nominal horse power of engine 
and |/140= 11.83 

1'2 = constant multiplier 



Answer 14*19 = diameter of waste water pipe in inches. 



Example 2. — What is the diameter of waste water pipe, in 
enches, of an engine whose nominal horse power is 885 ? 

385 = nominal horse power of engine 
and |/385 = 19*62 

1*2 = constant multiplier 



Answer 23*54 = diameter of waste water pipe in inches. 



CAPACITY OF THE FEED PUMP. 

The relative volumes of steam and water are at 15 lbs. on the 
-square inch, or the atmospheric pressure, 1,669 to 1 ; at 30 lbs., oi 



PROPER DIMENSIONS OF THE FEED PUMP. 225 

15 lbs. on the square inch above the atmospheric pressure, 881 to 
1 ; at 60 lbs., or 45 lbs. above the atmospheric pressure, 467 to 1 ; 
and at 120 lbs., or 105 lbs. above the atmospheric pressure, 249 to 1. 
In every engine, taking into account the risks of leakage and 
priming in the boiler, the feed pump should be capable of dis- 
charging twice the quantity of water that is consumed in the 
generation of steam ; and in marine boilers it is necessary to 
blow out as much of the supersalted water as the quantity that 
is raised into steam, in order to keep the boiler free from saline 
incrustations. But if this water is discharged by leakage or 
priming, the object of preventing salting is equally fulfilled. 
Pumps, especially if worked at a high rate of speed, do not fill 
themselves with water at each stroke, but sometimes only half 
fill themselves, and sometimes do not even do that. Then in 
steam vessels, one pump should be able to supply both engines 
with steam, and the pump is generally only single-acting, while 
the cylinder is double-acting. If, therefore, we wish to see 
what size of pump we ought to supply to an engine in which 
the terminal elasticity of steam in the cylinder is equal to the 
atmospheric pressure, we know that the quantity of water in 
the steam is just TeV^th of the volume of the steam ; but as we 
require to double the supply to make up for waste, the volume 
of water supplied will on this ground be ^Vs- > an ^ as the pump 
may only half fill itself every stroke, the capacity of the pump 
must on this ground be i-gVg- of the volume of steam. But then 
the pump is only single-acting, while the cylinder is double-act- 
ing, on which account the capacity of the pump must be doubled, 
in order that it may in a half stroke discharge the water re- 
quired to produce the steam consumed in a whole stroke. This 
would make the capacity of the pump 16 8 69 , or ^-J-g- of the capa- 
city of the cylinder, and a less proportion than this is inadvisa- 
ble in the case of marine engines. Even with this proportion, 
one feed pump would not supply all the boilers, as it ought to 
be able to do in case of accident happening to the other, unless 
it should happen that the pump draws itself full of water at 
each stroke instead of half full, as it will nearly do if the mo- 
tion of the engine is slow and the passages leading into it large, 



226 PROPORTIONS OP STEAM-ENGINES, 

and if at the same time the valves are large and have not much 
lift. In the case of engines working at a high speed, -^^ of the 
capacity of the cylinder for the capacity of the feed pump is 
scarcely sufficient, especially if there be no air vessel on the 
suction side of the pump, which in such pumps should always 
be introduced. In the engines of the 'Clyde,' 'Tweed,' 'Tay,' 
and ' Teviot,' by Messrs. Caird, the feed pump is -^f-oth of the 
capacity of the cylinder. In steam vessels there is no doubt 
always the resource of the donkey engine to make up for any 
deficiency in the feed. But it is much better to have the main 
feed pumps of the engine made of sufficient size to compensate 
for all the usual accidents befalling the supply of feed water. 
Of course, the supply of feed water required will vary mate- 
lially with the amount of expansion with which the steam is 
worked, and also with the amount of superheating ; and in the 
old flue boilers with the chimney passing up through the steam 
chest, there was always a considerable degree of superheating. 
A rule applicable to all pressures of steam and to moderate rates 
of expansion is as follows : — 

TO FIXD THE PEOPEE CAPACITY OF THE FEED PUMP. 

Rule. — Jlultiply the capacity .of tJie cylinder in cubic inches by 
the total pressure of the steam in the boiler on each square 
inch {or by tlie load on each square inch of the safety valve 
plus 15 lbs. on each square inch for the pressure of the at- 
mosphere), and divide the product by 4,000. The quotient 
is the proper capacity of the feedpump in cubic inches when 
the pump is single-acting and the engine is double-acting. 

If the pump should be double-acting, one-half of the above 
capacity will suffice. 

Example 1. — What is the proper volume of the working part 
of the plunger of an engine with a 74-inch cylinder and 7-§-feet 
stroke, the steam in the boiler being 5 lbs. per square inch above 
the atmospheric pressure ? 

The area in square inches of a circle 74 inches diameter is 
±,300, which, multiplied by 7i feet or 90 inches, gives 387,000 



COLD WATER PUMP. 227 

cubic inches as the capacity of the cylinder. Now if the steam 
in the boiler be 5 lbs. per square inch above the atmosphere, it 
will have a total pressure of 5 -f 15, or 20 lbs. per square inch. 
Multiplying, therefore, 387,000 by 20, we get 7,740,000, which, 
divided by 4,000, gives 1,935 as the proper capacity of the feed 
pump in cubic inches. If now the stroke of the pump be 51 
inches, we divide 1,935 by 51, which gives us 38 inches nearly 
as the proper area of the feed pump plunger. This area corre- 
sponds to a diameter of 7 inches, which is a better proportion 
than that subsisting in the engines of the ' Clyde,' ' Tweed,' 
1 Tay,' and ' Teviot,' which, with a 74-inch cylinder, 7-§- feet 
stroke, and 51 inches stroke of pump, had the feed pump .plung- 
ers of only 6 inches diameter. 

Example 2. — "What is the proper volume of the working part 
of the plunger of a locomotive feed pump, having cylinders of 
18 inches diameter and 2 feet stroke, working with a pressure 
of 85 lbs. pressure above the atmosphere ? 

The area of a circle 18 inches diameter is 254*5 square inches, 
which, multiplied by 24 inches, which is the length of the 
stroke, gives 6,108 cubic inches as the capacity of the cylinder. 
If the steam be 85 lbs. above the atmosphere, then the total press- 
ure must be 100 lbs. per square inch, and 6,108x100=610,800, 
which, divided by 4,000, gives 152*7 as the capacity of the feed 
pump in cubic inches. This is a somewhat larger proportion of 
feed pump than is usually given in locomotive engines. In the 
locomotive ' Iron Duke ' the diameter of the feed pump plunger 
is 2t inches and the stroke 24 inches. But 152*7 divided by 24 
inches gives an area of 6*36 square inches, which answers to a 
diameter of plunger of 2£ inches. In locomotives, however, as 
in marine engines, the feed pumps are very generally made too 
small, so that the proportion given in the rule appears prefera- 
ble to that commonly adopted. 

COLD-WATER PUMP. 

The proper dimensions of the cold-water pump can easily be 
determined by a reference to the number of cubic inches of wa- 
ter, at a given temperature, that are required to condense a 



223 PROPORTIONS OP STEAM-ENGINES. 

cubic inch in the form of steam. There is no need, however, 
of going through the details of the process, and the proper di- 
mensions of the pump will be found by the following rule : — 

TO DETEEMINE THE PEOPEE DLMEXSI03TS OF THE COLD-WATEE 

PUMP. 

JvUiE. — Multiply the square of the diameter of the cylinder in 
inches by the length of the stroke in feet, and divide the 
product by 4,400. The quotient is the proper capacity of 
the cold-water pump in cubic feet. 

Example 1. — What is the proper capacity of the cold-water 
pump in an engine, having a 60-inch cylinder and a 5|~feet 
stroke ? 

Here 60 x 60 = 3,600, which multiplied by 5£ is 19,800, and 
this divided by 4,400 is 4*5, which is the proper capacity of the 
cold-water pump in cubic feet. 

Example 2. — "What is the proper capacity of the cold-water 
pump in the case of an engine, with a 2-feet cylinder and 3-feet 
stroke ? 

Here 24 x 24 = 576, and this multiplied by 3 = 1,728, which 
divided by 4,400 = '39 cubic feet, or multiplying -39 by 1,728, 
we get the capacity in cubic inches, which is 673*92. This is a 
somewhat larger content than is sometimes given in practice. 
Maudslay's 16-horse land engine has a 24-inch cylinder and 
3-feet stroke, and the cold-water pump has a diameter of 6| 
inches, and a stroke of 18 inches, which gives a capacity of 594 
cubic inches, instead of 673, as specified above. The larger di- 
mension is the one to be preferred. 

FLY-WHEEL. 

Boulton and Watt's rule for finding the sectional area of the 
fly-wheel rim is as follows : — 

Rule, — Multiply 44,000 times the length of the stroke in feet 
by the square of the diameter of the cylinder in inches, and 
divide the product by the square of the number of revolu- 
tions per minute, multiplied, by the cube of the diameter of 



PROPER DIMENSIONS OF THE FLY-WHEEL. 229 

the fly-wheel in feet. Tlie resulting number will oe tlie 
proper sectional area of the fly-wheel rim in square inches. 

Example. — What will be tlie proper sectional area of tlie 
fly- wlieel rim in square inches in the case of an engine, with a 
cylinder 24 inches diameter and 5 feet stroke, the fly-wheel be- 
ing 20 feet diameter. 

Here 44,000 multiplied by 5, which is the length of the 
stroke in feet, is 220,000. The square of the diameter of the 
cylinder in inches is 576, and 220,000 x 576 =126,720,000. The 
engine will make about 21 revolutions, the square of which is 
441, and the cube of the diameter of the fly-wheel in feet is 
8,000, which multiplied by 441 is 3,528,000. Finally 126,720,000 
divided by 3,528,000 is 3 5 '8, which is the proper area in square 
inches of the section of the fly-wheel rim. 

In an engine constructed by Mr. Oaird, with a 24-inch cylin- 
der, 5-feet stroke, and 20-foot fly-wheel, the width of the rim 
was 10 inches, and the thickness 3f inches, giving a sectional 
area of 37*5 square inches, which is somewhat larger than Boul- 
ton and Watt's proportion. 

Suppose that we take the sectional area in round numbers at 
36 square inches, and the circumference of the fly-wheel or 
length of rim if opened out at 62 feet or 744 inches, then there 
will be 36 times 744, or 26,784 cubic inches of cast iron in the 
rim, or dividing by 1,728, we shall have 15*5 cubic feet of cast 
iron. But a cubic foot of cast iron weighs 444 lbs. Hence 15|- 
cubic feet will weigh 6,882 lbs., and this weight revolves with a 
speed of 21 times 62, or 1,303 feet per minute, or 21*7 feet per 
second, or 260*4 inches per second. To find the height in 
inches from which a body must have fallen, to acquire any given 
velocity in inches per second, we square the velocity in inches, 
and divide the square by 772*84, which gives the height in 
inches. Now the square of 260*4 is 67,808, which divided by 
772*84 = 87 inches, or 7£ feet, so that the energy treasured in 
the fly-wheel is equal to a weight of 6,882 lbs. falling through 
7J feet, or to a weight of 49,984*5 lbs. falling through 1 foot. 
Now the area of the cylinder being in round numbers 452 
square inches, the total pressure upon it, if we allow an effec- 



230 PROPORTIONS OP STEAM-ENGINES. 

tive pressure including steam and vacuum of 7 lbs. per square 
inch, as was the proportion allowed in "Watt's engines, will be 
3,164 lbs., and the length of stroke being 5 feet, we shall have 
3,164 lbs. moved through 5 feet, or 5 times this, which is 15,820 
lbs. moved through 1 foot in each half stroke of the engine. 
Dividing now 49,984*5 foot-pounds, the total power resident in 
the fly-wheel at its mean velocity, by 158*20 foot-pounds, which 
is the power developed in each half stroke of the engine, we 
get 8 1 as the resulting number, which shows that there is over 
three times the power resident in the fly-wheel that is devel- 
oped in each half stroke of the engine. In cases where great 
equability of motion is required, this power of fly-wheel is not 
sufficient, and in some engines, the proportion is made six times 
the power developed in each half stroke, or, in other words, the 
fly-wheel is twice as heavy as that computed above. 

GOVERNOR. 

The altitude of the height of the cone in which the arms re- 
volve, measuring from the plane of revolution to the centre of 
suspension, will be the same as that of a pendulum which makes 
the same number of double beats per minute that the governor 
makes of revolutions ; or if the number of revolutions per minute 
be fixed, and we wish to obtain the proper height of cone, we 
divide the constant number 375*36 by twice the number of revo- 
lutions, which gives the square root of the height of the cone ; 
and, consequently, the height itself is equal to the square of this 
number. These relations are exhibited in the following rules : — 

TO DETEEMIXE THE SPEED AT WHICH A GOVEEXOB MTST BE 
DEIVEX, WHEX THE HEIGHT OF THE COXE IS FIXED IX WHICH 
THE AEMS EEVOLVE. 

Rule. — Divide the constant number 375*36 by twice the square 
'root of the height of the cone in inches. The quotient is the 
proper number of revolutions per minute. 

Example. — A governor with arms 30|- inches long, measuring 
from the centre of suspension to the centre of the ball, revolves 



PROPER PROPORTIONS OP THE GOVERNOR. 231 

u: the mean position of the arms at an angle of about 30 degrees. 
with the vertical spindle forming a cone about 26J inches high. 
At what number of revolutions per minute should this governor 
be driven ? 

Here the height of the cone being 26*5 inches, the square root 
of which is 5*14, and twice the square root 10*28, we divide 
375*36 by 10*28, which gives us 36*5 as the proper number of 
revolutions at which the governor should be driven. 

TO DETEEMIXE THE HEIGHT OF THE COISE IN WHICH THE AEMS 
MUST EEVOLVE, WHEN THE VELOCITY OF EOTATIOIST OF THE 
GOVEENOE IS DETEEMINED. 

Rule. — Divide the constant number 375*36 by twice the number 
of revolutions which the governor makes per minute, and 
square the quotient, which will be the height in inches which 
the cone icill assume. 

Example. — Suppose that a governor be driven with a speed 
of 86-J- revolutions per minute, what will be the height of the 
cone in which the balls will necessarily revolve, measuring from 
the centre of suspension of the arms to the plane of revolution 
of the balls? 

Here 36*5 x 2 = 73, and 375*36 divided by 73 =5*14, and 
5*14 squared is equal to 26*4196, or very nearly 26*5 inches, 
which will be the height of the cone. 

"When the arms revolve at an angle of 45 degrees with the 
spindle, or at right angles with one another, the centrifugal force 
is equal to the weight of the balls ; and when the arms revolve 
at an angle of 30 degrees with the spindle, they form with the 
base of the cone an equilateral triangle. 



STRENGTHS OF LOW-PRESSURE LAND ENGINES. 

PISTON ROD. 

The piston rod is made one-tenth of the diameter of the 
cylinder, except in locomotives, where it is made one-seventh 



232 PROPORTIONS OF S1EAJI-ENGINES. 

of the diameter. The piston rod is sometimes made of steel, or 
of iron converted into steel for a certain depth in. This enables 
it to acquire and maintain a better polish than if made of iron. 

MAIN LINKS. 

The main links are the parts which connect the piston rod 
with the beam. They are usually made half the length of the 
stroke, and their sectional area is 113th the area of the piston. 

AIR-PUMP ROD. 

The diameter of the air-pump rod is commonly made one- 
tenth of the diameter of the air-pump. 

BACK LINKS. 

The sectional area of the back links is made the same as that 
of the air-pump rod. 

END STUDS OF THE BEAM. 

The end studs of the beam are usually made the same diam- 
eter as the piston rod. Sometimes they are of cast-iron, but 
generally now of wrought. The gudgeons of water wheels are 
generally loaded with about 500 lbs. for every circular inch of 
then.' transverse section, which is nearly the proportion that ob- 
tains in the end studs of engine beams. But the main centre is 
usually loaded beyond this proportion. 

MAIN CENTRE. 

The strength of this part will be given in the strengths ol 
marine engines. But when of cast-iron it is usually made about 
one-fifth of the diameter of the cylinder. 

In a cylinder of 2-1 inches diameter this will be 4'8 inches, or 
gay 4f inches ; and this proportion of strength will be about nine 
times the breaking weight, if we suppose the main centre to be 
overhung as in marine engines. Thus, in a cylinder of 24 inches 
diameter, and, consequently, of 452 square inches area, the total 
load on the piston with 20 lbs. on each square inch is 9,040 lbs, 



PROPER DIMENSIONS OF TF/E MAIN BEAM. 233 

But as the strain at the main centre is doubled from the beam 
acting as a lever of 2 to 1, it follows that the strain at the main 
centre will be 18,080 lbs. The ultimate tensile strength of com- 
mon cast-iron being 12,000 per square inch of section, and the 
tensile and shearing strength being about the same, |th of 12,000, 
or 1,338 lbs., will be the proper load to place on each square inch 
of section ; and 18,080 divided by 1,333 will give the proper sec- 
tional area in square inches, which will be 13| square inches 
nearly. This area corresponds to a diameter of a little over 4| 
inches. But the strength is virtually doubled by the circum- 
stance of the main centre of land engines being supported at 
both ends. 

MAIN BEAM. 

The rules in common use for proportioning the main beams 
of engines are the same as those which existed prior to Mr. 
Ilodgkinson's researches on the strength of cast-iron girders, 
which showed that the main element of strength was the bot- 
tom flange. But as in the case of engine beams the strain is 
alternately up and down, the top and bottom flanges, or beads 
of the beam, require to be of equal strength. Cast-iron is a bad 
material for engine beams, unless the central part be made of 
open work of cast-iron, and the edge of the beam be encircled 
by a great elliptical or lozenge-formed hoop, as is done in some 
of the American engines. But if the beam be made wholly of 
cast-iron, a much larger proportion of the metal should be col- 
lected in the top and bottom flanges than is at present the ordi- 
nary practice. 

The usual length of the main beam is three times the length 
of the stroke ; the usual breadth is equal to the diameter of the 
cylinder, and the usual mean thickness is -j-^th of the length. 
The rule is as follows : — 

TO FIND THE TEOPEE DIMENSION'S OF THE MAIN BEAM OF A 

LAND ENGINE. 

Rule. — Divide the weight in lbs. acting at the centre by 250 and 
multiply the quotient by the distance between the extreme cen- 
tres. To find the depth, the breadth being given : Divide iht 



234 PROPORTIONS OF STEAM-ENGINES. 

product by tlie breadth in indies, mid extract the square root 
of the quotient, which is the depth. 

The depth, of the beam at the ends is usually made one-third 
of the depth at the middle. 

It will be preferable, howevei, to investigate a rule on the 
basis of Mr. Hodgkinson's rule for proportioning cast-iron gird « 
ers, which is as follows : 

Multiply the sectional area of the "bottom flange in inches by 
tJie depth of the beam in inches, and divide the product by the 
distance between the supports also in inches, and 514 times the 
quotient will be the breahing weight in cicts. 

If the breaking weight be expressed in tons, the constant 
number 514 must be divided by 20, which gives the breaking 
weight as 25*7, or say 26 tons, whereas experiment has shown 
that if the flange were to be formed of malleable iron instead 
of cast, the breaking weight would not be less than 80 tons ; or, 
in other words, that with the same sectional area of flange, the 
beam would be more than three times stronger. 

It is a common practice in the case of girders to make the 
strength equal to three times the breaking weight when the load 
is stationary, and to six times the breaking weight when the 
(oad is movable. But these proportions are too small, and less 
than nine or ten times the breaking weight will not give a suf- 
ficient margin of strength in the case of engines where the mo- 
tion is so incessant, and where heavy strains may be accidentally 
encountered from priming or otherwise. In the case of an en- 
gine, the weight answering to the breaking weight is the load 
on the piston ; and if we suppose the fly-wheel to be jammed, 
and the piston to be acting with its full force to lift or sink the 
main centre, it is clear that the strain on the main centre, and, 
therefore, od the beam, will be equal to twice the strain upon 
the piston, since the beam acts under such circumstances as a 
lever of 2 to 1. The problem we have now to consider is how 
many times the working weight must be less than the breaking 
weight to give a sufficient margin of strength in any given beam 
or, in other words, what proportions must the beam have to 
possess adequate working strength. 



PROPER DIMENSIONS OF THE MAIN BEAM. 235 

To take a practical example from an engine in constant work. 
The engine with a cylinder of 24 inches diameter has a main 
beam 15 feet (or 180 inches) long ; 30 inches deep in the middle ; 
and with a sectional area of flange of 7 square inches.. The 
breaking weight of such a beam in cwts. will be 7 x 30 x 514 
divided by 180 = 600 cwt. nearly, and this multiplied by 112 lbs. 
= 67,200 lb., which is the breaking weight in pounds avoirdu- 
pois. The area of the cylinder in round numbers is 452 square 
inches ; but as there is a leverage of 2 to 1, this is equivalent to 
an area of cylinder of 904 square inches set under the middle of 
the beam and pulling it downwards, the beam being supposed 
to be supported at both ends. Dividing now 67,200 by 904 we 
get the pressure per square inch on the piston that would break 
the beam, which is a little over 74 lbs. per square inch of the 
area of the piston, or 58 lbs. per circular inch. If we suppose 
the working pressure of steam on the piston to be 6*27 lbs. per 
circular inch, or 7*854 lbs. per square inch, then the working 
strength of the beam will be about 9} times its breaking strength, 
which would give an adequate margin for safety. But if we 
suppose the working pressure to be 12*54 lbs. per circular inch, 
or 15 '718 lbs. per square inch, the working strength would in- 
such case be only about 4| times the breaking strength, and the 
beam would be too weak. 

The strength of a cast-iron beam of any given dimensions 
varies directly as the sectional area of the edge flange ; or, if 
the sectional area of that flange be constant, the strength of the 
beam varies directly as the depth, and inversely as the length. 
If while the sectional area of the flange remains the same the 
depth of the beam is doubled without altering the length, then 
the strength is doubled. But if the length be also doubled, the 
strength remains the same as at first. As the length of an en- 
gine-beam is doubled when we double the length of the stroke, 
and as in any symmetrical increase of an engine when we double 
the length of the stroke we also double the diameter of the cyl- 
inder, to which the depth of the beam is generally made equal, 
large beams with the same area of flange, and made in the ordi- 
nary proportions, would be as strong as small beams, except that 



236 pbopobtions of sihax-ksgixz - . 

the load increases as the square of the diameter of the cylinder, 
and consequently the area of the edge flange must increase in 
the same proportion. These considerations enable ns to fix the 
following role for the strength of main beams : — 



j — - — _i 

?. dik — li bt :'-: depth of the "beam, equal to the diameter cfthe 
cylinder \ and the length of the oeam equal to three times the 
length of the stroke. Then to find Vie area of the edge 
flange: Multiply the area of the cylinder in square inches 
oy the total pressure of steam and vacuum on each square 
inch of the piston, and divide the product oy 650. The 
quotient is the proper area of the flange of the oeam in 

Z zmple 1. — "Wl if is Hie proj at sectional area of the flange 
of the main beam of an engine, with cylinder 24 inches diam- 
eter and 5-feet stroke, the pressure on the piston being 20 lbs. 
~ rr = " z:.:r — :'_ : 

Here the area of the cylinder will be 452 inches, which mul- 
tiplied by 20 gives 9,040, and dividing by 650 we ge: 1 \ \ square 
inches, which is the proper sectional area of the ed^e bead or 
::i: : :: ._-. : -■ ■".." , 

Izample 2. — What is the proper sectional area of the flange 
of the main beam of an engine with a ejfindei 50 inches di- 
ameter, 1 _ 1 feel stroke, and with a pressure of steam on the pis- 
ton of 20 lbs. per square inch 1 

The area of a cylinder 60 inches diameter is 2,824 square 
inches, and 2,824 multiplied by 20=56,480, which divided by 
650=8T square inches nearly. Such a flange, therefore, if 14§ 
inches broad, would be 6 inches thick. The beam would be 
5 feet deep at the middle, and 87f feet long between the ex- 



PROPER DIMENSIONS OF THE MAIN BEAM. 237 

ANOTHER EULE FOE FINDING THE SECTIONAL AREA OF EACJH 
EDGE FLANGE OF THE MAIN BEAM. 

Rule. — Multiply the diameter of tlie cylinder in inches oy one- 
tliird of the length of the stroke in inches, and oy the total 
pressure on each square inch of the piston, and divide the 
product oy 650. The quotient is the proper sectional area 
in square inches of each flange or dead on the edge of the 
oeara. 

Example 1. — What is the proper sectional area of the flange 
on the edge of the main heam of an engine with a 24-inch cylin- 
der, 20 lbs. total pressure on piston per square inch, and 5 feet 
stroke ? 

Here 24 x 20 (which is one-third of the stroke in inches) x 20 
(the pressure of the steam and vacuum per square inch) = 9,600, 
which divided by 650=14 , 'T sq. in., which is the area required. 

Example 2. — What is the proper sectional area of the flange 
on the edge of the main beam of an engine with a 60-inch cylin- 
der, 12|-feet stroke, and with a pressure on the piston of 20 lbs. 
per square inch ? 

Here 60 x 50 (which is one-third of the stroke in inches) x 20 
(the pressure of the steam per square inch) = 6,000, which di- 
vided by 650 gives 92 as the sectional area of the edge bead in 
square inches. Such a flange, if 15^- inches broad, would be 
6 inches thick. These results it will be seen are very nearly 
the same as those obtained by the preceding rule ; and one in- 
ference from these rules is that nearly all engine beams are at 
present made too weak. The purpose of the web of the beam 
is mainly to connect together the top and bottom flanges, so that 
there is no advantage in making it thicker than suffices to keep 
the beam in shape ; with which end, too, stiffening feathers, both 
vertical and horizontal, should be introduced upon the sides of 
the beam. The first cast-iron beams were made like a long 
hollow box to imitate wooden beams, and this form would still 
be the best, unless an open or skeleton beam, encircled with 
a great wrought-iron hoop after the American fashion, be 
adopted. 



238 peopobtiohs of bteam-engihes. 

COJTNECTING-ROD. 

The connecting-rods of land engines are now usually made 
of wr ought-iron, and when so made, the proportions will "be the 
same, or nearly so, as those given under the head of marine en- 
gines. "When made of cast-iron the configuration is such that 
the transverse section at the middle assumes the form of a cross, 
this form being adopted i'j give greyer lateral stillness. The 
length of the rod is usually made the same as the length of the 
beam, namely, three times the length of the stroke, and the 
area of the cross section of the rod at the middle is commonly 
made ^th of the area :: the cylinder, and the sectional area at 
the ends A-th oi the area of the cylinder. Such a strength is 
needlessly great, and is quite out of proportion to the strength 
commonly given to the beam. Thus, in the case of an engine 
with a 21-inch cylinder, the area of the piston is 452 square 
inches ; and if we fcak 3 1 3 lbs. per square inch as the load on the 
piston, then the total load on the piston will be 9,010 lbs. If 
the working load be made ^th of the breaking load, as in the 
of the beam, then the breaking load should be 81,360 lbs., 
and the strength of the connecting-rod should be such that it 
would just break with that load on the piston. Xovr the tensile 
strength of the weakest cast-iron is about 12.000 lbs. per square 
inch of section, while its crushing strength is about five times 
that amount. Dividing 81,361 lbs., the total tensile strength of 
the rod, by 12,000. the tensile strength of one square inch, we 
get about 7 square inches as the proper area of the smallest part 
of the connecting-rod when of cast-iron. But -g^th of 452 (which 
is the area of the cylinder in square inches) is 13 square inches, 
from all of which it follows that while the main beams of en- 
gines are commonly made, too weak, the cast-iron connecting- 
r : Is are commonly made too strong. This, however, is partly 
done for the purpose of balancing the weight of the piston and 
its connections. 

rLY-TVHEEL SHAJT. 

The fly-wheel shaft of land engines is usually made of cast- 
iron. The following is the rule on which such shafts are usually 

proportioned : — 



PROPER DIAMETER OF FLY-WHEEL SHAFT. 239 

TO FIND THE DIAMETER OF THE FLY-WHEEL SHAFT AT SMALLEST 
PART, "WHEN IT IS OF CAST-IRON. 

Rule.— Multiply the square of the diameter of the cylinder in 
inches by the length oftlie crank in inches ; extract the cube 
root of the product ; finally multiply the result ~by *3025. 
The product is the diameter of the fly-wheel shaft at the 
smallest part in inches. 

Example 1. — "What is the proper diameter of the fly-wheel 
shaft, when of cast-iron, in the case of an engine with a diameter 
of cylinder of 64 inches and a stroke of 8 feet ? 

64 = diameter of the cylinder in inches 
64 



4096 = square of the diameter 
48 = length of crank in inches 



196608 
58-15 = ^/196608 and 5815 x -3025 = IY'59, which is the proper 
diameter of the fly-wheel shaft at the smallest part. 

Example 2. — What is the proper diameter, at the smallest 
part, of the cast-iron fly-wheel shaft of an engine, with a diameter 
of cylinder of 40 inches, and 5 feet stroke ? 

40 = diameter of cylinder in inches 
40 



1600 — square of diameter of cylinder 
30 — length of crank in inches 



48000 
36-30 = ^48000 and 36:30 x '3025 == 10-98, which is the proper diara. 
eter of the shaft in inches. 

MR. WATT'S RULE FOR THE NEOKS OF HIS CRANK SHAFTS. 

RirLE. — Multiply the area of the piston in square inches by the 
pressure on each square inch (and which Mr. Watt toolc at 
12 lbs.), and by the length of the cranio in feet. Divide the 
product by 81*4, and extract the cube root of the quotient, 
which is the proper diameter of the shaft in inches. 



2 1 PBOPOBHOV i OS STJBAM-S K S IKS S ■ 

.7: imple 1. — TThat is the proper diameter of the fly-wheel 
shaft in an engine, with a cylinder 64 inches diameter and 8 feet 

The area of a cylinder 64 inches diameter is 8.217 square 
inches, which multiplied by 13 = 38,604, and this multiplied by 
4, which is the length of the crank in feet, is 154,4:16. This 
divided by 31/4 = 4. 17 ■". :he cube root of which is 17*01 

inches. 

Example 2. — "What is the right diameter, according to Mr. 
Waifs r^le, of the fly-wheel shaft of an engine, with a 24-inch 
cylinder, 5 feet stroke, .and with a pressure of 12 lbs. on each 

~_e ;r:.\ :: :'_e :y'±iier is 47- = ;-:.re inzTir?, ~ "_::"_ in^rl- 
plied by 12 = 5,424, and this multiplied by 2^, which is the 
length of the crank in fe;: = 13,560, which divided by 31*4 = 
431, the cube root of which is 7i inches, which is the proper 
diameter of the shaft. In Mr. a engine the diameter is 8 



TO FT5TD THE PEOPEB THICKNESS OF THE T, KEGE EYE OF THE 
:r.-^N~K FOE FLY-WHEEL SHAFT. W HK.\ OF CAST-IBOH. 

Rule. — Multiply the square of the length of the crank in inches 
by 1*561, and then multiply the square of the diameter of the 
cylinder in inches by "1335/ multiply the sum of these prod- 
ucts by the square of the diameter of the cylinder in inches; 
divide this product by 666*283/ divide this quotient by the 
length of the crani in inches ; finally extract the cube root of 
the quotient. The result is the proper thickness of the large 
eye of crank far fly-wheel shaft in inches, when of cast-iron. 

JT :i\~ 1. — ?.e::±rei tie 7/r:;:er tZiiciness :: :'_: I:.r_:e ;-~r 
of crank for fly-wheel shaft, when of cast-iron, of an engine 
whose length of stroke is 8 feet, and diametci ;:' -Under 64 

ill I'll f 5. 



THICKNESS OP LARGE EYE OF CRANK. 241 

48 — length of crank in inches 
48 

2304 = square of length of crank in inches 
1-561 = constant multiplier 

8596-5 

64 = diameter of cylinder in inches 
64 

4096 = \ S( l uare °f diameter of cylinder 
( in inches 

•1235 = constant multiplier 

505-8 
3596-5 



4102-3 = sum of products 

if the d 
linder in inches 



4096 = \ S( l uare of the diameter of the cy- 



dMsor 111 } = 666-283)16803020-8 

Length of crank = 48)25219-045 

525*397 

and ^/525-397 = 8-07 which is the proper thickness of the large eye of 
the crank in inches, when of cast-iron. 

Example 2.— Required the proper thickness of the large eye 
of the crank for fly- wheel shaft, when of cast-iron, of an en- 
gine, whose length of stroke is 5 feet, and diameter of cylinder 
40 inches. 



30 = length of crank in inches 
30 

900 = square of length of crank in inches 
1-561 = constant multiplier 

1404-9 



11 



242 PROPORTIONS OP STEAM-ENGINES. 

40 = diameter of cylinder in inches 
40 

1600 

•1235 = constant multiplier 



197*6 
1404-9 

1602-5 = sum of products 

1600 = square of diameter of cylinder 



dr^sor Dt [ ~ 666 " 283 ) 2564 000-0 
Length of crank = 30 inches 3848*2 

' 128-3 
and ^128-3 = 5 -04 inches is the proper thickness in this engine of the 
large eye of the crank, when of cast-iron. 

TO FIND THE PEOPEE BEEADTH OF THE WEB OF THE CEANK AT 
THE CENTRE OF THE FLY-WHEEL SHAFT, WHEN OF CAST-IEON, 
SUPPOSING THE BEEADTH TO BE CONTINUED TO THE CENTEE 
OF THE SHAFT. 

Eule. — Multiply the square of the length of the crank in inches 
by 1*561, and then multiply the square of the diameter of the 
cylinder in indies by *1235 ; multiply the square root of the 
sum of these products by the square of the diameter of the 
cylinder in inches ; divide the product "by 23*04, and finally 
extract the cube root of the quotient. The final result is the 
breadth of the crank at the centre of the fly-wheel shaft y 
when the crank is of cast-iron. 

Example 1. — What is the proper breadth of the wen of the 
crank at the centre of fly-wheel shaft, when of cast-iron, in the 
case of an engine, with a diameter of cylinder of ,64 inches, and 
length of stroke 8 feet ? 

48 ■=. length of crank in inches 

48 

2304 = square of length of crank 
1*56] = constant multiplier 

8596-5 



PROPER BREADTH OF WEB OF CRANK. 213 

64 = diameter of cylinder in inches 
64 



4096 = square of diameter of cylinder 
•1235 = constant multiplier 

505-8 
3596-5 



4102-3 = sum of products 



v/4102-3= 64-05 nearly 

4096 = square of diameter of cylinder in inches. 

23-04)262348-80(11395-34 
2304 



3214 
2304 

9108 
6912 



21968 
2073% 

12320 
11520 

8000 
6912 



10880 
9216 

1664 



"$11395-34 = 22-5 inches, which is the proper breadth of the web of the 
crank, when of cast-iron, supposing the breadth to be continued to the 
centre of the fly-wheel shaft. 

Example 2.— "What is the proper breadth of the web of a 
cast-iron crank at the centre of the fly-wheel shaft (snpposing it 
to be so far extended), in the case of an engine with 40 inches 
diameter of cylinder and 5 feet stroke ? 



244 PROPORTIONS OF STEAM-ENGINES. 

SO = length of crank in inches 
SO 

900 = sqnare of length of crank in inches 
1*561 = constant multiplier 



1404*9 



40 = diameter of cylinder in inches 
40 

1600 = square of diameter of cylinder 
•1235 = constant multiplier 



197-6 

1602*5 = sum of products 
^1602*5 = 40*3 nearly 
1600 



23*04)644S0*0 



2798*6 nearly 
^793-6 = 14*09, which is the proper breadth in inches of a cast iron 
crank in an engine of this size, supposing the taeadth to be continued 
to the fly-wheel shaft. 

TO nXD THE PEOPES THICKNESS OP THE WEB 07 A CAST-LEON 
CEANK AT THE CESTEE OF THE FLY-WHEEL SHAFT. 

Bomj.— Multiply the square of the length of the crank in inches 

by 1-561, and then multiply tTce square of the diameter of the 
cylinder in inches by '1235 ; multiply the square root of the 
sum of these products ly the square of the diameter of the 
cylinder in incites; divide the product ly 1*32; finally 
extract the cute root of the quotient. The result is the proper 
thickness of the web of a cast-iron cranlc in inches at the cen- 
tre of the fly-wheel shaft, supposing the thklcness to be ex- 
tended to that point. 

Examvle 1.— Bequired the proper thickness of the web of a 
^ast-iron crank a* the centre of the fly-wheel shaft (supposing it 



PROPER THICKNESS OF WEB OF CRANK. 245 

to be so far extended), in the case of an engine with 64 inches 
diameter of cylinder, and 8 feet stroke. 

48 == length of crank in inches 
48 



2304 = square of the length of crank 
1*561 = constant multiplier 



3596-5 



64 = diameter of cylinder in inches 
64 



4096 = square of diameter of cylinder 
•1235 = constant multiplier 



505-8 
3596-5 



4102*3 == sum of products 
and ^4102-3 = 64*05 nearly 

4096 = square of diameter of cylinder 



Constant 
divisor 



I ~ 184-32)262348-5 



1422-33 
and <ty 1423-33 = 11*25 

Example 2. — "What is the proper thickness ot the web of a 
cast-iron crank at centre of fly-wheel shaft (supposing it to be so 
far extended), in the case of an engine with 40 inches diameter 
of cylinder, and 5 feet stroke ? 

30 = length of crank in inches 
30 

900 _ j square of length of crank in 
— ( inches 

1-561 = constant multiplier 



1404*9 



246 pbopoktions or steam-engines. 

40 = diameter of cylinder m inches 
40 

1600 = square of diameter of cylinder 
•1235 =: constant multiplier 



197-6 

1404-9 



16025 

4 1602-5 = 40-3 nearly 
1600 



Octant I = m . S2)5tts(H) 

S49-8 
and .y349"8 = *7 - 04, "which is the proper thickness in inches of the wd> 
of a cast-iron crank for this engine, measuring at the centre of the fly- 
wheel shaft 

CRANK PET. 

The crank pins of land engines havinj sast-ivon Hanks, are 

generally made of cast-iron, and are in diameter about one-sixth 
of the diameter of the cylinder. 

MILL GEARING. 

Bonlton and Watt, by whom the present system of iron 
gearing was introduced, proportioned their wheels on the follow- 
ing consideration : — ' That a bar of cast-iron 1 inch square and 
12 inches long, bears 600 lbs. before it breaks ; 1 inch long will 
bear 7,200 lbs., and ^th of this = 480 Ibs^ is the load which 
should be put on the wheel,' for each square inch in section of 
the tooth. Bonlton and Watt's rule for the strength of geared 
wheels is consequently as follows : — If H = the actual horses 
power which the wheel has to transmit ; d, the diameter of the 
wheel in feet, and r, the revolutions of the wheel per minute ; 
then H x S06 

-j — = the strength, and the strength divided by Hw 

breadth in inches =jp\ or the square of the pitch in inches. 



PROPER PROPORTIONS OF TOOTHED WHEELS. 247 



3ence H = *' x l * d x r and p =a/ R X 806 , which equations 
306 V bx dxr 

put into words are as follows : 

TO FIND THE NUMBEE OF ACTUAL HOESES POWEE WHICH A GIVEN 
WHEEL WILL TEANSHIT, ACCOEDING TO BOULTON AND WATT'S 
PEAOTIOE. 

Rule. — Multiply the square of the pitch in inches by the breadth 
of the wheel in inches, by its diameter in feet, and by the 
number of revolutions it maJces per minute, and divide the 
product by the constant number 306. The quotient is the 
number of actual horses' 1 power which the wheel will safely 
transmit, according to Boulton and Watt's practice. 

TO FIND THE PEOPEE PITCH OF A WHEEL LN INCHES TO TEANS- 
MIT A GIYEN POWEE, ACCOEDING TO BOULTON AND WATT'S 
PEACTIOE. 

T\U~LE.—Multip)ly the breadth of the teeth in inches by the diam- 
eter of the wheel in feet, and by the number of revolutions it 
makes per minute, and reserve the product as a divisor. Next 
multiply the number of actual horses' power which the wheel 
has to transmit by the constant number 306, and divide the 
product by the divisor found as above. Finally, extract the 
square root of the quotient, which is the proper pitch of the 
wheel in inches, according to Boulton and Watt 1 s practice. 

Instead, however, of reckoning the strain in horses' power, it 
is preferable to reckon it as a pressure or weight applied to the 
acting tooth of the driving wheel. If t = the thickness of the 
tooth in inches, w = the pressure upon it in lbs., and c a con- 
stant multiplier, which for cast-iron is *025, for brass, *035, and 
for hard wood, *038, then t = c «J w, by which formula we can 
easily find the proper thickness of the tooth, and twice the 
thickness of the tooth with the proper allowance for clearance, 
gives the pitch. This formula put into words is as follows : — 



24:8 PROPORTIONS OF STEAM-ENGINES. 

TO FLXD THE PEOPEE THICKNESS OF TOOTH OE A CAST-IBOU 
"WHEEL TO TEANS3JIT WITH SAFETY ANT GIVEN PEESSTJEE. 

Rule. — Multiply the square root of the pressure in pounds act- 
ing at the pitch line oy the constant nuniber *025. The 
product is the proper thickness of the tooth in inches. 

Example 1. — "What is the proper thickness of the teeth of a 
cast-iron wheel moved by a pressnre of 233*33 lbs. at the pitch 
circle ? 

Here 7 233-33 = 15 *27, and this mnltiplied by -025 = -381, 
which is the proper thickness of the teeth in inches. 

Example 2. — What is the proper thickness of the teeth of a 
cast-iron wheel which is moved round by a pressnre of 46,666*6 
lbs. at the pitch circle ? 

It will be easiest to solve this question by means of logarithms. 
As the index of the logarithm is always one less than the number 
of places above unity filled by the number of which the logarithm 
has to be found ; and as there are five such places in the number 
46,666*6, it follows that the index of the logarithm will be 4, and 
the rest of the logarithm will be found by looking for the nearest 
number to 46,666*6 in the tables, and which number will be 
4.666, the logarithm answering to which is 668945. The residue 
Q-Q, however, has not yet been taken into account, and to include 
it we must multiply the number found opposite to the logarithm 
in the column marked D, commonly introduced in logarithmic- 
tables (and which is a column of common differences), by the 
number we have not yet reckoned, namely, 6*6 ; and cut off a 
number of figures from the product equal to those in the mul- 
tiplier, adding the residue to the logarithm, which will thereupon 
become the correct logarithm of the whole quantity. The com- 
mon difference in this case is 93, which multiplied by Q'Q gives 
613*8, and cutting off the 3*8 we add the 61 to the logarithm 
already found, which then becomes 4*669006. Dividing this by 
2, we get 2*334503, which will be the logarithm of the number 
that is the square root of 46,666*6. As the index of the loga- 
rithm is 2, there ^vill be three places above unity in the number, 
ind looking now in the logarithm tables for the number answer- 



PROPER PROPORTIONS OF TOOTHED WHEELS 249 

ing to the logarithm nearest 334503, we get the number 216, the 
logarithm of which is 334454. The number 216 is consequently 
the square root of 46,6G6'6 very nearly, as to extract the square 
root by logarithms, we have ouly to divide the logarithm of the 
number by 2, and the number answering to the new logarithm 
thus found will be the square root of the original number. Now 
216 multiplied by *025 = 5*400, which consequently is the thick- 
ness in inches of each of the teeth of this wheel. 

GENERAL EULES EEGAEDING GEAEING. 

The pitch should be in all cases as fine as is consistent with 
the required strength. "When the velocity of the motion exceeds 
3-£ feet per second, the larger of the two wheels should be fitted 
with wooden teeth, the thickness of which should be a little 
greater than that of the iron teeth. The breadth of the teeth in 
the direction of the axis varies very much in practice. But 
where the velocity does not exceed 5 feet per second, a breadth 
of tooth in the line of the axis equal to four times the thickness 
of the tooth will suffice. This is nearly the same thing as a 
breadth equal to twice the pitch. "Where the velocity at the 
pitch circle is greater than 5 feet per second, the breadth of the 
teeth should be 5 times the thickness of tooth, the surfaces being 
kept well greased. But if the teeth be constantly wet, the 
breadth should be 6 times the thickness of tooth at all velocities. 
The best length of the teeth is fths of the pitch, and the length 
should not exceed fths of the pitch, and the effective breadth 
of the teeth should not be reckoned as exceeding twice, the 
.ength ; any additional breadth being good for wear, but not for 
strength. In the Soho practice the length of the teeth is made 
T 4 2 -ths of the pitch outside, and -j^ths of the pitch inside of the 
pitch circle, the whole length being -^ths or fths of the pitch. 
The London practice is to divide the pitch into 12 parts, and to 
adjust the length of the tooth by allowing -j^ths without, and 
^jths within the pitch circle, the entire length of tooth being 
j'gths of the pitch. The projection of the teeth beyond the pitch 
circle will be \th of the pitch, and the surface in contact between 
11* 



250 PROPORTIONS OF STEAM-ENGINES. 

the teeth of the two wheels will be half the pitch. About -£th 
of the pitch should be left unoccupied at the bottom of the teeth 
for clearance. 

With regard to the least number of teeth that is admissible in 
the smaller of two wheels working together, 12 to 18 teeth will 
answer well enough in crane work, where a pinion is employed 
to give motion to a wheel at a low rate of speed. But for quick 
motions, a pinion driven by a wheel should never have less than 
from 30 to 40 teeth. 

The best form of teeth is the epicycloidal, and in general the 
proper curve is obtained by roiling a circle of wood carrying a 
pencil on another circle of wood answering to the pitch circle, 
the point of the tooth being described by the rolling circle trav- 
ersing the outside of the pitch line, and the root by traversing 
the inside of the pitch line. The diameter of the rolling circle 
should be 2'22 times the pitch. Some teeth are not epicycloi- 
dal, but the roots are radii of the pitch circle, and the points 
are described with compasses from the pitch centre of the next 
tooth. 

In the following table will be found the thickness and pitch 
of teeth answering to different amounts of load or pressure at the 
pitch circles. But it may here be remarked that such large 
pitches as 12 and 13 inches are practically not used. In cases 
where such large pressures are to be transmitted as answer to 
pitches over 5 inches or thereabout, it is usual to distribute the 
load by placing two or more parallel wheels upon the same shaft, 
working into corresponding pinions ; and it is also usual to set 
the teeth of each wheel a little in advance of the teeth of the 
wheel next it, so as to divide the pitch, and thus render the 
action of the teeth smoother and more continuous. 



EXAMPLES OP HEAVY GEARING. 



251 



PEOPOETIOXS OF THE TEETH OF CAST-IEON WHEELS. 



' 


Pitch of 






Pitch of 




I Pressure in lbs. 


teeth in inches, 


Thickness 


Pressure in lbs. 


teeth in inches, 


Thickness 


at the 


allowing 


of teeth in 


at the 


allowing 


of teeth in 


pitch circle. 


one-tenth for 
clearance. 


inches. 


pitch circle. 


one-tenth for 
clearance. 


inches. 


233-33 


•79S 


•38 


11666-65 


5-6705 


2-7002 


, 349-95 


•981 


•467 


13999 -9S 


6-2118 


2-9580 • 


466-66 


1-134 


•540 


16333-31 


6-7099 


3-1952 


1 5S3-32 


1-263 


•604 


18666-64 


7-1728 


3-4156 


699-99 


1-333 


•661 


20999-97 


7-6079 


3-6228 ' 


816-65 


1-5 


•716 


23333-3 


8-0194 


3-8188 ' 


933-32 


1-604 


•763 


25666-63 


8-4109 


40052 


1049-98 


1-7 


•809 


27999-96 


8-7848 


4-1832 


1166-65 


1-793 


•854 


30333-29 


9-1470 


4-3557 


1233-31 


1-88 


•895 


32666-62 


9-4S87 


4-5184 


1399-98 


1-964 


•935 


34999-95 


9-8218 


4-6770 


1516-64 


2-044 


•973 


37333-28 


10-1439 


4-S304 


1633-31 


2-121 


1-04 


39666-61 


10-4560 


4-9790 


1749-97 


2-196 


1-045 


41999-94 


10-7592 


5-1234 


1866-64 


2-263 


1-08 


44333-27 


11-0540 


5-2638 


19S3-3 


2-338 


1-113 


46666-6 


11-3412 


5-4006 


2099-97 


2-405 


1-145 


49999-93 


11-7381 


5-5896 


2216-63 


2-47 L 


1-177 


52338-26 


12-0103 


5-7192 


2333-3 


2-538 


1-203 


54666-59 


12-2749 


5-8452 


2449-96 


2-598 


1-237 


56999-92 


12-5341 


5-9686 


2566-63 


2-659 


1-266 


59333-25 


12-7883 


6-0S97 


2683-29 


2-720 


1-295 


60666-58 


12-9310 


6-1576 


2799-96 


2-777 


1-322 


62999-91 


13T773 


6-2749 


4666-66 


3-5S6 


1-7078 


65333-24 


13-3S93 


6-3759 


699999 


4-3924 


2-0916 


67666-57 


13-6566 


6-5031 


9333-32 


5-0719 


2-4152 


69999-99 


13-8901 


6-6143 



It will be useful to illustrate the application of these rules to 
the case of heavy gearing by one or two practical examples. 

In a steamer with engines by Messrs. Penn and Son there 
are two cylinders of 82J inches diameter and 6 feet stroke, 
giving motion to a toothed wheel 14 feet diameter consisting of 
four similar wheels bolted together, the. teeth being 12 inches 
broad and 5*86 inches pitch. The area of a cylinder 82i inches 
being 5,346 square inches, there will be a total pressure on the 
piston — if we reckon the mean average pressure upon each square 
inch at 25 lbs. — of 133,650 lbs. But as there are two pistons, the 
total pressure on the two pistons will be 267,300 lbs. Now the 
diameter of the geared wheel being 14 feet, its circumference 
will be 44 feet, and as at each movement of the pistons up and 
down through the length of the stroke, or through a distance. 
of 12 feet, the wheel makes one revolution, or moves through 44 
feet, the pressure at the circumference of the wheel will be less 



252 PEOPORTIOXS OF STEAM-EXGIXES. 

than that on the pistons in the proportion in •which 44 exceeds 
12, so that by multiplying 267,300 by 12 and dividing the product 
by 44 we get the equivalent or balancing pressure at the circumfer- 
ence of the wheel, and which is 69, 673 lbs. As, however, this load 
is distributed among four wheels, there will only be one-fourth of 
69,673, or 17,418 lbs. to be borne by each of them. According 
to the rule we have given, therefore, the square root of 17,418 
multiplied by "025 will be the proper thickness of each tooth in 
inches. Xow VI 7,418 = 132, and 132 x -025 = 3*3, which by our 
rule is the proper thickness of the tooth in inches, and twice this, 
or 6"6, with one-tenth or '3 for clearance, will be the pitch = 6*9, 
whereas the actual pitch is 1 inch less than this. If the multi- 
plier be made # 02, instead of '025, the value obtained will agree 
more nearly with this example, as 132 x '02 = 2 '64, which will be 
the thickness of tooth, and 2'64 x 2 ■ = 5 "28, to which adding ^th 
of the thickness of the tooth for clearance, or # 264, we get 5-544 
inches as the pitch. If we take the pressure at 20 lbs. per square 
inch on the pistons instead of 25 lbs., then the total pressure on 
the two pistons will be 213,840 lbs., which reduced to the equiv- 
alent pressure at the periphery of the wheel will be 58,320 lbs. 
The fourth of this is 14,580, the logarithm of which is 4*163758, 
the half of which is 2*081879, the natural number answering to 
which is 120 - 7, which multiplied by -025 = 3'1175, winch is the 
proper thickness of the tooth in inches for this amount of strain. 
It will be seen, therefore, that the strength which our rule gives 
is somewhat greater than that of this example. 

Let us now take an example by a different maker, and we 
select the geared engines of the steamer ' City of Glasgow,' con- 
structed by ilessrs. Tod and Macgregor. There were two cylin- 
ders in this vessel, each 66 inches diameter and 5 feet stroke, and 
the motion was communicated from the crank shaft to the screw 
shaft by means of four parallel wheels, 7 feet diameter, 8 inches 
broad, and 4 inches pitch. The area of a cylinder 66 inches 
diameter is 3,421 square inches, and the area of two such cylin- 
ders will, consequently, be 0.842 square inches. If we take the 
pressure urging the pistons at 20 lbs. per square inch, the total 
Dressure on the pistons will be 136,840, which reduced to the 



EXAMPLES OF HEAVY GEARING. 253 

pressure at the periphery of the wheel — which moves 2*2 times 
faster than the pistons — will be 62,200 lbs. ; and as the pressure 
is divided among four wheels there will be one-fourth of 62,200, 
or 15,550 lbs. on each. The logarithm of this number is 4*191430, 
the half of which is 2*095715, the natural number answering 
to which is 124*7, and 124*7 multiplied by *025 = 3*1175, which 
is half as much again as the actual strength given in these 
wheels. 

We may take still another example, and shall select the caso 
of the Tire Queen,' a screw yatch constructed by Messrs. 
Robert Napier and Sons. In this vessel there are two cylinders, 
each of 36 inches diameter and 36 inches stroke, and the motion 
is communicated from the crank shaft to the screw shaft through 
the medium of three parallel wheels 8£ f ee * diameter placed on 
the end of the crank shaft. The pitch of the teeth is 3*55 inches 
and two of the wheels are 4 inches broad, and one of them 6 
inches broad. The two narrow wheels may be reckoned as equiv- 
alent to one broad one, so we may consider the strain to be 
divided between two wheels. The area of each cylinder is 
1,018 square inches, and if we reckon two cylinders of this area, 
with a pressure of 20 lbs. per square inch, urging the piston of 
each, the total pressure urging the pistons will be 40,720 lbs. 
The double stroke of the piston is 6 feet, and the circumference 
of the wheel is 26*7 feet ; and as the wheel revolves once while 
the pistons are making a double stroke, the relative velocities will 
be 6 and 26*7, and the relative pressures 26*7 and 6. Multiply- 
ing, therefore, 40,720 by 6 and dividing by 26*7, we get 9,150 lbs. 
as the pressure at the circumference of the wheel ; and as this 
load is to be divided between two wheels, there will be a load 
of 4,575 lbs. upon each. The logarithm of 4,575 is 3*660391, 
the half of which is 1*830195, the natural number answering to 
which is 67*64, which multiplied by *025 gives 1*691 as the proper 
thickness of tooth in this wheel. Twice 1*691 is 3*382, to which 
if we add -j^th of the thickness of the tooth, or *169 for clear- 
ance, we get 3*55 as the proper pitch of this wheel, and this is 
the very pitch which is really given. In this case, therefore 
the rule and the example perfectly correspond. The rule give? 



254 PROPORTIONS OF STEAM-ENGINES. 

sufficient strength to represent the mean thickness of wooden 
and iron teeth — the wooden teeth being a little thicker, and the 
iron teeth a little thinner than the amount which the rule pre 
scribes. 



MARINE ENGINES. 

The rules which I have given in my "Catechism of the 
Steam-Engine " for fixing the proper proportions of the parts of 
marine engines, take into account the pressure of the steam with 
which the engine works. But in order that the proportions thus 
arrived at may be more easily comparable with the proportions 
subsisting in the engines of different constructors, in which 
the pressure is assumed as tolerably uniform, it will be more 
convenient so to frame the rules that a uniform pressure of 25 
lbs. per square inch of the area of the piston shall be supposed to 
be at all times existing. In cases where it is desired to ascer- 
tain the dimensions proper for a greater pressure than 25 lbs., it 
will be easy to arrive at the right result by taking an imaginary 
cylinder of as much greater area than the real cylinder as the 
real pressure exceeds the assumed pressure of 25 lbs., and then 
by computing the strengths and other proportions as if for this 
imaginary cylinder, they will be those proper for the real cylin- 
der. Thus if it be desired to ascertain the strengths proper for 
an engine with a cylinder of 30 inches diameter, and with a 
pressure on the piston of 100 lbs. on the square inch, the end 
will be attained if we determine the strengths proper for an 
engine of 60 inches diameter, and with 25 lbs. pressure on the 
square inch ; for the area of the larger cylinder being four times 
greater than that of the smaller, the same total force will be ex- 
erted with one-fourth of the pressure. So, in like manner, if it 
be wished to ascertain the strengths proper for an engine with a 
cylinder GO inches diameter, and with a pressure on the piston 
of 50 lbs. per square inch, we shall find them by determining 
the proportions suitable for an engine with an area of piston 
twice greater than the area of a piston 30 inches diameter, and 
jv r bich avea will be that answering to a diameter of 42-g- inches. 



DIMENSIONS OP THE CROSSHEAD. 255 

By this mode of procedure a table of proportions adapted to the 
ordinary pressures will be made available for determining the 
proportions suitable for all pressures, as we have only to fix upon 
an assumed cylinder which shall have as much more area as the 
intended pressure has an excess of pressure over 25 lbs. per 
square inch, and the proportions proper for this assumed cylin- 
der will be those proper for the real cylinder with the pressure 
intended. In this way the strengths fixed for marine engines 
may also be made applicable to locomotives and to high and low 
pressure engines of every kind. In the following rules, there- 
fore, it will be understood the strengths and other proportions 
are those proper to an assumed pressure on the piston, including 
steam and vacuum, of 25 lbs. per square inch, and the computa- 
tions are for side lever engines, but for the most part are appli- 
cable to all kinds of engines. 



CROSSHEAD. 

TO FUID THE PEOPER THICKNESS OF THE WEB OE THE CEOS8- 
HEAD AT THE MIDDLE. 

Rule. — Multiply the diameter of tlie cylinder in inches oy '072. 

Example 1. — Let 40 inches be the diameter of the cylinder. 

Then 40 inches x '072 = 2*880 inches, which is the proper 
thickness of the web of the crosshead at the middle in this en- 
gine. 

Example 2. — Let 64 inches be the diameter of cylinder. 

Then 64 inches x *072 = 4*608 inches, which is the proper 
thickness of the web of the crosshead at the middle in this en- 
gine. 

TO FIND THE PEOPEE THICKNESS OF THE WEB OF THE OEOSS- 
HEAD AT THE JOUENAL. 

Rule. — Multiply the diameter of the cylinder in inches by *061. 

Example 1. — Let 40 inches be the diameter of the cylinder. 
Then 40 inches x *061 = 2*440 inches, which is the proper 



25G PROPORTIONS OF STEAM-ENGINES. 

thickness of the web of the crosshead at the journal in this en 
gine. 

Example 2. — Let 64 inches be the diameter of the cylinder. 

Then 64 inches x '061 = 3*904 inches, which is the proper 
thickness of the web of the crosshead at the journal in this en- 
gine. 

TO FLXD THE PEOPEE DEPTH OP TEE WEB OF THE CEOSSHEAP 

AT THE MIDDLE. 

Rule. — Multiply the diameter of the cylinder in inches oy "268 

Example 1. — Let 40 inches be the diameter of the cylinder. 
Then 40 inches x *26S = 10*720 inches, which is the propei 
depth of the web of the crosshead at the middle in this engine. 

Example 2. — Let 64 inches be the diameter of the cylinder. 
Then 64 inches x '268 = 17*152 inches, which is the proper 
depth of the web of the crosshead at the middle in this engine. 

TO FIND THE PEOPEE DEPTH OF THE WEB OF THE CEOSSHEAD 

AT JOTJENALS. 

Rule. — Multiply the diameter of the cylinder in inches oy *101. 

Example 1. — Let 40 inches be the diameter of cylinder. 

Then 40 inches x *101 = 4*040 inches, which is the proper 
depth of the web of the crosshead at journals in this engine. 

Example 2. — Let 64 inches be the diameter of the cylinder. 

Then 64 inches x *101 •= 6*464 inches, which is the proper 
depth of the web of the crosshead at journals in this engine. 

TO FIXE THE PEOPEE DIAMETEE OF THE JOTJEXALS OF THE CEOSS- 
HEAD. 

Rule. — Multiply the diameter of the cylinder in inches oy '086. 

Example 1. — Let 40 inches be the diameter of the cylinder. 
Then 40 inches x *086 = 3*440 inches, which is the propei 
<1 ameter of the journals of the crosshead in this engine. 
Example 2. — Let 64 inches be the diameter of cylinder. 



DIMENSIONS OF THE CROSSHEAD. 257 

Then 64 inches x *086 = 5-504 inches, which is the proper 
diameter of the journal of crosshead in this engine. 

TO FEND THE PEOPEE LENGTH OF THE JOURNALS OF THE CEOSS« 

HEAD. 

The length of the journals of the crossheads should he equal 
to ahout 1£ times their diameter, hut on the whole it appears to 
be advisable to mate the journals of the crosshead as long as 
they can he conveniently got. 

TO FIND THE PEOPEE THICKNESS OF THE EYE OF THE 
CEOSSHEAD. 

Rule.— Multiply the diameter of the cylinder in inches dy *041. 

Example 1. — Let 40 inches he the diameter of the cylinder. 

Then 40 inches x '041 = 1*640 inches, which is the proper 
thickness of the eye of the crosshead in this engine. 

Example 2. — Let 64 inches he the diameter of the cylinder. 

Then 64 inches x '041 = 2-624 inches, which is the proper 
thickness of the eye of the crosshead in this engine. 

TO FIND THE PEOPEE DEPTH OF THE EYE OF THE CEOSSHEAD. 

Rule. — Multiply the diameter of the cylinder in inches uy -286. 

Example 1. — Let 40 inches he the diameter of the cylinder. 

Then 40 inches x '286 = 11-440 inches, which is the proper 
depth of the eye of the crosshead in this engine. 

Example 2. — Let 64 inches be the diameter of the cylinder. 

Then 64 inches x '286 = 18-304 inches, which is the proper 
depth of the eye of the crosshead in this engine. 

TO FE*D THE PEOPEE DEPTH OF GLBS AND CUTTEE PASSING 
THEOTJGH THE CEOSSHEAD. 

Rule. — Multiply the diameter of the cylinder in inches ~by *105, 



258 PROPORTIONS OF STEAM-ENGINES. 

Example 1. — Let 40 inches be the diameter of the cylinder. 

Then 40 inches x *105 = 4*200 inches, which is the proper 
depth of the gibs and cntter passing through the crosshead in 
this engine. 

Example 2. — Let 64 inches be the diameter of the cylinder. 

Then 64 inches x *105 = 6*720 inches, which is the proper 
depth of the gibs and cutter passing through the crosshead in 
this engine. 

TO EIND THE PEOPEE THICKNESS OF THE GIBS AND CUTTER 
PASSING THEOUGH THE CEOSSHEAD. 

Rule. — Multiply the diameter of the cylinder in inches ly '021. 

Example 1. — Let 40 inches be the diameter of the cylinder. 

Then 40 inches x "021 = *840 inches, which is the proper 
thickness of the gibs and cutter passing through the crosshead 
hi this engine. 

Example 2. — Let 64 inches be the diameter of the cylinder. 

Then 64 inches x *021 = 1*344 inches, which is the proper 
thickness of the gibs and cutter passing through the crosshead 
in this engine. 



SIDE RODS. 

TO FIND THE PEOPEE DIAMETEE OF THE OTLINDEE SIDE EODS 
AT THE ENDS. 

Rule. — Multiply the diameter of the cylinder in inches ~by *065. 

Example 1. — Let 40 inches be the diameter of the cylinder. 
Then 40 inches x '065 = 2*6 00 inches, which is the proper 
diameter of cylinder side rods at ends in this engine. 

Example 2. — Let 64 inches be the diameter of the cylinder. 
Then 64 inches x "065 = 4*160 inches, which is the proper 
diameter of the cylinder side rods at ends in this engine. 

The diameter of the side rods at the middle should be about 



DIMENSIONS OF THE SIDE RODS. 259 

one-fourth more than the diameter at the ends. Thus a side rod 
5 inches diameter at the ends will be 6 J inches diameter at the 
middle. 

The area of the horizontal section of iron through the middle 
of eye of side rod is usually about one-half greater than the sec- 
tional area of the side rod at ends. 

TO FIND THE PEOPER BREADTH OF THE BUTT OF THE SIDE ROD 

IN INCHES. 

Rule. — Multiply the diameter of the cylinder in inches ay *077. 

Example 1. — Let 40 inches be the diameter of the cylinder. 

Then 40 inches x '077 = 3 '080 inches, which is the proper 
breadth of butt of side rod in this engine. 

Example 2. — Let 64 inches be the diameter of the cylinder. 

Then 64 inches x '077 = 4*928 inches, which is the proper 
breadth of butt in this engine. 

TO FIND THE PEOPER THICKNESS OF THE BUTT OF THE SIDE 

RODS. 

Rule. — Multiply the diameter of the cylinder in inches oy *061. 

Example 1. — Let 40 inches be the diameter of the cylinder. 
Then 40 inches x *061 = 2*440 inches, which is the proper 
thickness of the butt of the side rod in this engine. 

Example 2. — Let 64 inches be the diameter of the cylinder. 
Then 64 inches x *061 = 3*904 inches, which is the proper 
thickness of the butt of the side rod in this engine. 

TO FIND THE PROPER MEAN THICKNESS OF THE STRAP OF THE 
SIDE ROD AT THE CUTTER. 

Rule. — Multiply the diameter of the cylinder in inches oy *032. 

Example 1. — Let 40 inches be the diameter of the cylinder. 

Then 40 inches x *032 = 1*280 inches, which is the proper 
mean thickness of the strap of side rod at the cutter in this 
engine. 



2'Jj Fz:?:zr::ys or szr kv-zs-riyzs. 

Example 2. — Let 64 indies be the diameter of the cylinder. 
Then 64 inches x "032 = 2*048 inches, which is the proper 
mean thickness* of the strap of side rod at Hie cotter in this 



ZF '. Z SCZ3 



Bulb. — Multiply the diameter of the cylinder in, inches "by "023. 

Zz-zrtz".<: 1. — Let -'.' inches : e the 1 izcze :e: itthe ijlinie:-'. 

Izez =i iz;zcs hi =-r£ inches, — hich is the 7: zzer zczz 
thickness of Hie strap of the side rod below the cotter in this 
enzhze. 

Example 2. — Let 64 inches be the diameter of the cylinder. 

Then 64 inches x "023 = 1*4^2 inches, which is the proper 
Zr.,i :'-' ~"-" -. ; . ; :c :zr sec;. - :: the 5: It : : 1 '.. el:~ the iztter in 

this cZZZCe. 

z: fztz zzcz e?z:zzz zzzztt :r czz 2: aczz crcctr. :t 

stzz z:z. 

P.zzz. — __T ": _ \. :" : '.',: : :::' .'/the :; h :.h ' .' : .' :hz 1 ; h?. 

Zzz :\'-: 1. — Let -'. inches he the iizzceter ct the cylinder. 
Then i! inches =: '15 = I'll inches., vrhich i» hie trie :t 



__cn :'-■ zzcz cs x -.^ = ■;■■__ mines, tt_;^. 1= — e tr:: :: 
Letch :c zi.s 11: iztter :: :" sile r: 2 in this czzize. 

z: :r: zzz zzirzz zzz:zz~ss :r zczs -lco czzzzz :z 

dma. — Multiply the diameter of the cylinder in inches fty "016. 

Z :' ".'.'■: 1. — het =~. inches *:e the lizzzeter :c the lyzhler. 

Lzez i! izzhes x hi: = ■•;4 inches, — hi:h is the 'tc'vr 
zzichness :t zhs :nl cntter cf site r: i in this ez.hze. 

Z .:. :::'.. L — L-: 14 inches eziL the ihzzzecer :: ijhzter. 

Then B4 inches X "016 = 1"02 inches, — hi;h is the props 
uii:hzes.s :: zi; s czl czcter :: rile r: i in this enzizc. 



DIMENSIONS OE THE PISTON KOD. 261 

PISTON ROD. 

TO FIND THE PEOPEE DIAMETEE OF THE PISTON EOD. 

Rule. — Divide the diameter of the cylinden in -inches oy 10. 

Example 1.— Let 40 inches "be the diameter of the cylinder. 

Then 40 inches -f- 10 = 4'0 inches, which is the proper diame- 
ter of piston rod in this engine. 

Example 2. — Let 64 inches he the diameter of the cylinder. 

Then 64 inches -f- 10 = 6*4 inches, which is the proper diame- 
ter of piston rod in this engine. 

TO FLND THE PEOPEE LENGTH OP THE PAET OF THE PISTON EOD 

IN THE PISTON. 

Rule. — Divide the diameter of the cylinder in inches ~by 5. 

Example 1. — Let 40 inches be the diameter of the cylinder. 

Then 40 inches -J- 5 = 8*0 inches, which is the proper length 
of the part of the piston rod in the piston in this engine. 

Example 2. — Let 64 inches he the diameter of the cylinder. 

Then 64 inches -i- 5 = 12*8 inches, which is the proper length 
of the part of the piston rod in the piston in this engine. 

TO FLND THE MAJOB DIAMETEE OF THE PAET OF THE PISTON 
EOD LN THE PISTON. 

Rule. — Multiply the diameter of the cylinder in inches 1y *14. 

Example 1. — Let 40 inches equal the diameter of cylinder. 

Then 40 inches x *14 = 5*60 inches, which is the proper 
major diameter of the part of the piston rod in piston in this 
engine. 

Example 2. — Let 64 inches he the diameter of the cylinder. 

Then 64 inches x *14 = 8*96 inches, which is the proper 
major diameter of the part of the piston rod in piston in this 
eugine. 

TO FLND THE MLNOE DIAMETEE OF THE PAET OF THE PISTON 
EOD IN THE PISTON. 

Rule. — Multiply the diameter of the cylinder in inches hy *115. 



262 PROPORTIONS OF STEAM-ENGINES. 

Example 1. — Let 40 inches be the diameter of the cylinder. 

Then 40 inches x *115 = 4*600 inches, which is the proper 
minor diameter of the part of the piston rod in piston in this 
engine. 

Example 2. — Let 64 inches be the diameter of the cylinder. 

Then G4 inches x *115 =7*360 inches, which is the proper 
mi j i or diameter of the part of the piston rod in piston in this 
engine, 

TO FIND THE 3IAJOE DIAMETER OF THE PART OF THE PISTON 
EOD IN THE CEOSSHEAT). 

Rule. — Multiply the diameter of the cylinder in inches oy '095. 

Example 1. — Let 40 inches be the diameter of the cylinder. 

Then 40 inches x *095 = 3*800 inches, which is the proper 
major diameter of the part of the piston rod in the crosshead 
in this engine. 

Example 2. — Let 64 inches be the diameter of the cylinder. 

Then 64 inches x '095 = 6*080 inches, which is the proper 
major diameter of the part of the piston rod in the crosshead in 
this engine. 

TO , FIND THE MINOE DLOLETEE OF THE PAET OF THE PISTON 
EOD LN CEOSSHEAD. 

Rule. — Multiply the diameter of the cylinder in inches ~by *09. 

Example 1. — Let 40 inches be the diameter of the cylinder. 

Then 40 inches x *09 = 3*60 inches, which is the proper 
minor diameter of the part of the piston rod in crosshead in this 
engine. 

Example 2. — Let 64 inches be the diameter of the cylinder. 

Then 64 inches x *09 '== 5*76 inches, which is the proper 
minor diameter of the part of the piston rod in crosshead in this 
engine. 

TO FEND THE PEOPEE DEPTH OF THE CUTTEE THEOUGH PISTON. 

Rule. — Multiply the diameter of the cylinder in inches oy *085. 
Example 1.— Let 40 inches be the diameter of the cylinder. 



DIMENSIONS OP THE CONNECTING-ROD. 263 

Then 40 inches x *085 = 3*400 inches, which is the propel 
depth of the cutter through the piston in this engine. 

Example 2. — Let G4 inches he the diameter of the cylinder. 

Then 64 inches x *085 = 5*440 inches, which is the proper 
depth of the cutter through the piston ir this engine. 

TO FIND THE PEOPEE THICKNESS OF THE CUTTEE THROUGH 

PISTON. 

Rule. — Multiply the diameter of the cylinder in inches oy *035. 

Example 1. — Let 40 inches by the diameter of the cylinder. 

Then 40 inches x *035 = 1*400 inches, which is the proper 
thickness of cutter through the piston in this engine. 

Example 2. — Let 64 inches be the diameter of the cylinder. 

Then 64 inches x *035 = 2*240 inches, which is the proper 
thickness of cutter through piston in this engine. 



CONNECTING-ROD. 

TO FIND THE PEOPEE DIAMETEE OF THE CONNECTING-EOD AT 

THE ENDS. 

Rule. — Multiply the diameter of the cylinder in inches oy *095. 

Example 1. — Let 40 inches he the diameter of the cylinder. 

Then 40 inches x *095 = 3*800 inches, which is the proper 
diameter of the connecting-rod at the ends in this engine. 

Example 2. — Let 64 inches he the diameter of the cylinder. 

Then 64 inches x *095 = 6*080 inches, which is the proper 
diameter of the connecting-rod at the ends in this engine. 

The diameter of the connecting-rod at the middle will vary 
with the length, hut is usually one-fifth more than the diameter 
at the ends. Thus a connecting-rod 7*7 inches diameter at the 
ends will be 9*25 inches diameter at the middle. 

TO FESD THE MAJOE DIAMETEE OF THE PAET OF OONNECTING- 
EOD IN THE CEOSSTAIL. 

Rule. — Multiply the diameter of the cylinder in inches oy *098. 



264 PBOPOETIOXS OF STEAM-EXGIXES. 

Example 1. — Let 40 inches be the diameter of the cylinder. 

Then 40 inches x '098 = 3 # 920 inches, vrhich is the proper 
major diameter of the part of the connecting-rod in the cross 
tail in this engine. 

Example 2. — Let 64 inches he the diameter of the cylinder. 

Then 64 inches x "098 = 6*272 inches, vrhich is the proper 
major diameter of the part of connecting-rod entering the cross 
tail in this engine. 

TO riNT) THE PEOPEE 3EDTOE D IAM KEKB OF THE PAET OF COX 

XECTLXG-EOD EXTEELXG- TEE CEOSSTAEL. 

Rule. — Ifultiply the diameter of the cylinder in inclies ~by '09. 

Example 1. — Let 40 inches he the diameter of the cylinder. 

Then 40 inches x *09 = 3*60 inches, vrhich is the proper 
minor diameter of the part of the connecting-rod in the cross- 
tail in this engine. 

Example 2. — Let 64 inches he the diameter of the cylinder. 

Then 64 inches x *09 = 5*76 inches, vrhich is the proper 
minor diameter of the part of the connecting rod in the cross- 
tail in this engine. 

TO FTXD THE PEOPEE BEEADTH OF BUTT OF THE COXXECTLXG- 

EOD. 

Rule. — ifultiply the diameter of the cylinder in inches oy # 156. 

Example 1. — Let 40 inches he the diameter of the cylinder. 

Then 40 inches x "156 = 6 '240 inches, vrhich is the proper 
breadth of the bntt of connecting-rod in this engine. 

Example 2. — Let 64 inches be the diameter of the cylinder. 

Then 64 inches x *156 = 9'984 inches, vrhich is the proper 
breadth of the bntt of the connecting-rod in this engine. 

TO FLXD THE PEOPEE THICEXESS OF THE BUTT OF TEE COX- 

XECTTXG-EOD. 

Rule. — Bicide the diameter of the cylinder in inches oy 8. 

Example 1. — Let 40 inches be the diameter of the cylinder. 
Then 40 inches -f- 8 = 5*00 inches, vrhich is the proper thick- 
ness of the bntt of the connecting-rod in this engine. 



DIMENSIONS OF THE CONNECTING-ROD. 265 

Example, 2. — Let 64 inches be tlie diameter of the cylinder. 
Then 64 inches -~- 8 = 8*00 inches, which is the proper thick- 
ness of the butt of the connecting-rod in this engine. 

TO FINE SHE PEOPEE MEAN THICKNESS OF THE STEAP OF CON- 
NEC TESTG-EOD AT THE CUTTEE. 

Rule. — Multiply the diameter of trie cylinder in inches ~by '043. 

Example 1. — Let 40 inches be the diameter of the cylinder. 

Then 40 inches x *043 = 1*720 inches, which is the proper 
mean thickness of the connecting-rod strap at the cutter in this 
engine. 

Example 2. — Let 64 inches be the diameter of the cylinder. 

Then 64 inches x *043 = 2*752 inches, which is the proper 
mean thickness of the connecting-rod strap at the cutter in this 
engine. 

TO FIND THE PEOPEE MEAN THICKNESS OF THE CONNEGTING-EOD 
STEAP ABOVE CUTTEE. 

Bule. — Multiply the diameter of the cylinder in inches ~by *032. 

Example 1. — Let 40 inches be the diameter of the cylinder. 

Then 40 inches x "032 = 1*280 inches, which is the proper 
mean thickness of the connecting-rod strap above the cutter in 
this engine. 

Example 2. — Let 64 inches be the diameter of the cylinder. 

Then 64 inches X *032 = 2*048 inches, which is the proper 
mean thickness of the connecting-rod strap above the cutter in 
this engine. 

TO FIND THE PEOPEE DISTANCE OF CUTTEE FEOM END OF STEAP 
OF CONNECTING-EOD. 

Rule. — Multiply the diameter of the cylinder in inches hy *04S. 

Example 1. — Let 40 inches be the diameter of the cylinder. 

Then 40 inches x *048 — 1*920 inches, which is the proper 
distance of the cutter from the end of the strap of the connect- 
ing-rod in this engine. 

Example 2. — Let 64 inches be the diameter of the cylinder, 
12 



266 PPOPORTIONS OP STEAM-ENGINES. 

Then 64 inches x *048 = 3*072 inches, which is the propel 
distance of the cutter from the end of the strap of the connect- 
ing-rod in this engine. 

TO FIND THE PKOPER DEPTH OF THE GIBS AND CUTTEE PASSING 
THROUGH THE CEOSSTAIL. 

!£i;ie. — Multiply the diameter of the cylinder in inches by '105. 

Example 1. — Let 40 inches he the diameter of the cylinder. 

Then 40 inches X *105 = 4*20 inches, which is the proper 
depth of the gibs and cutter passing through the crosstail in this 
engine. 

Example 2. — Let 64 inches he the diameter of the cylinder. 

Then 64 inches x *105 = 6-720 inches, which is the proper 
depth of the gibs and cutter passing through the crosstail in this 
engine. 

The thickness of the cutters passing through the crosstail will 
be the same as the thickness of those passing through the cross- 
head. 

TO FIND THE PEOPER DEPTH OF THE GIBS AND OIJTTEE THEOUGH 
THE BUTT OF THE CONNECTING-EOD. 

Rule. — Multiply the diameter of the cylinder in inches by *11. 

Example 1. — Let 40 inches be the diameter of the cylinder. 

Then 40 inches x *11 == 4*40 inches, which is the proper 
depth of the gibs and cutter passing through the butt of the 
connecting-rod in this engine. 

Example 2. — Let 64 inches be the diameter of the cylinder. 

Then 64 inches x *11 = 7*04 inches, which is the proper 
depth of the gibs and cutter passing through the butt of the con- 
necting-rod in this engine. 

TO FIND THE THICKNESS OF THE GIBS AND CUTTEE THEOUGII 
THE BUTT OF THE CONNECTING-EOD. 

Rule.— Multiply the diameter of the cylinder in inches by *029. 

Example 1. — Let 40 inches be the diameter of the cylinder. 
Then 40 inches x *029 = 1*160 inches, which is the proper 



DIMENSIONS OF THE SIDE LEVER. 267 

thickness of the gibs and cutter passing through the butt of the 
connecting-rod in this engine. 

Example 2. — Let 64 inches be the diameter of the cylinder. 

Then 64 inches x*029 =1*856 inches, which is the proper 
thickness of the gibs and cutter passing through the butt of the 
connecting-rod in this engine. 

CROSSTAIL. 

The crosstail is made in all respects the same as the cross- 
head, except that the end journals, where the crosstail butts fit 
on, are made so that the length is only equal to the diameter of 
the journal, instead of being about 1^ times, as in the crosshead. 
But as the crosstail butts do not work on these journals or gudg- 
eons, but are keyed fast upon them, the shorter length is pre- 
ferable. The butts of the crosstail have the eyes nearly twice 
the diameter of the journals, or more accurately 1*8 times, and 
the butts for the reception of the straps for connecting to the 
side lever are made of the same dimensions as the butts of the 
side rods. 

SIDE LEYER AND STUDS OR CENTRES. 

The side lever is usually made of cast-iron. But it should be 
in all cases encircled by a strong wrought-iron hoop, thinned at 
the edge so that it maybe riveted or bolted all along to a flange 
cast on the beam for this purpose, and forming an extension of 
the usual edge bead. The proportions given in the rules are 
those of the common cast-iron side levers as usually constructed. 
But the strength will be increased three times if wrought-iron 
be substituted for cast in the top and bottom flanges or edge 
beads. 

TO FIND THE PEOPEE DEPTH OF THE. SIDE LEVE2 ACEOSS THE 

CEXTEE. 

Rule. — Multiply the length of the side lever in feet oy "7423 ; 
extract thecuoe root of the product and reserve the root for a 
multiplier. Then square the diameter of the cylinder in 



268 PROPORTIONS OF STEAM 'ENGINES. 

inches ; extract tlie cute root of the square. The product 
of the last result, and the reserved multiplier, is the depth of 
the side lever in inches across the centre. 

Example 1. — "What is the proper depth across the centre of 
the side lever in the case of an engine with a diameter of cylin- 
der of G4 inches and length of side lever of 20 feet ? 

Here 20 = length of side lever in feet 
•'7433 length of multiplier 



14-848 and 3/14,846 = 2-458 nearly 

Also 64 = diameter of cylinder 
64 



4096 and 3/4096 =16 

Hence depth at centre = 16 x 2-458 = 39*80 inches, or be- 
tween 39|- and 39 inches. 

Example 2. — "What is the proper depth across the centre of 
the side lever in the case of an engine with a diameter of cylin 
der of 40 inches, and length of side lever of 15 feet. 

Here 15 = length of side lever 
•'7423 



11-1345 and ^11-1345 = 2-232 

Also 40 = diameter of cylinder 
40 



1600 and ty 1600 = 11-69 which x 2-232 = 26-09, 
or a little over 26 inches. 

The depth of the side lever at the ends is determined by the 
depth of the eyes ronnd the end studs. The thickness of the 
side lever is usually made about -gVth of its length, and the 
breadth of the edge bead is usually made about -^ of the length 
of the lever between the end centres. 

TO FIND THE PBOPEE BIA1IETEE OF THE MAIN CENTRE JOTJENAL. 

Rule. — Multiply the diameter of the cylinder in inches' oy -183. 



DIMENSIONS OF THE SIDE LEVER. 269 

Example 1. — Let 40 inches be the diameter of the cylinder. 

Then 40 inches x *183 = 7'32 inches, which is the proper 
diameter of the main centre journal in this engine. 

Example 2. — Let 64 inches be the diameter of the cylinder. 

Then 64 inches x *183 = H'712 inches, which is the proper 
diameter of the main centre journal in this engine. 

TO FIND THE LENGTH OF THE MAIN CENTEE JOUENAL. 

Rule. — Multiply the diameter ofilie cylinder in inches 7>y '21 5. 

Example 1. — Let 40 inches be the diameter of the cylinder. 

Then 40 inches x '275 = 11*00 inches, which is the proper 
length of the main centre journal in this engine. 

Example 2. — Let 64 inches be the diameter of the cylinder. 

Then 64 inches x *275 = 17*60 inches, which is the proper 
length of the main centre journal in this engine. 

TO FIND THE DIAMETER OF THE END STUDS OF THE SIDE LEYEE. 

Rule. — Multiply the diameter of the cylinder in inches iy '07. 

Example 1. — Let 40 inches be the diameter of the cylinder. 

Then 40 inches x *07 = 2*80 inches, which is the proper 
diameter of the end studs of the side lever in this engine. 

Example 2. — Let 64 inches be the diameter of the cylinder. 

Then 64 inches x *07 = 4'48 inches, which is the proper 
diameter of the end studs of the side lever in this engine. 

TO FIND THE PEOPEE LENGTH OF THE END STUDS OF THE SIDE 

LEVEE. 

Rule. — Multiply the diameter of the cylinder in inches oy "076. 

Example 1. — Let 40 inches be the diameter of the cylinder. 

Then 40 inches x '076 = 3*04 inches, which is the proper 
.ength of the end of studs of the side lever in this engine. 

Example 2. — Let 64 inches be the diameter of the cylinder. 

Then 64 inches x *076 = 4*86 inches, which is the proper 
length of the end studs of the side lever in this engine. 

TO FIND THE PEOPEE DIAMETEE OF THE AIE-PUMP STUDS IN SILB! 

LEVEE. 

Rule. — Multiply the diameter of the cylinder in inches dy '045. 



270 PROPORTIONS OF STEAM-ENGINES. 

Example 1. — Let 40 inches be the diameter of the cylinder. 

Then 40 inches x '045 = 1*80 inches, which is the proper 
diameter of the stud in the side lever for working the air-pump 
of this engine. 

Example 2. — Let 64 inches be the diameter of the cylinder. 

Then 64 inches x "045 = 2*88 inches, which is the proper 
diameter of the air-pump studs in the side levers of this engine, 

TO FIND THE PEOPEB LENGTH OF THE AIE-PUMP STUDS SET IN TnE 

SIDE LEVEE. 

Rule.— -Multiply the diameter of the cylinder in inches oy *049. 

Example 1. — Let 40 inches be the diameter of the cylinder. 

Then 40 inches x *049 = 1*96 inches, which is the proper 
length of the air-pump studs in this engine. 

Example 2. — Let 64 inches be the diameter of the cylinder. 

Then 64 inches x *049 = 3*136 inches, which is the proper 
length of the air-pump studs in this engine. 

TO FUND THE PEOPEE DEPTH OF THE EYE BOUND THE END STUDS 

OF SLDE LEVEE. 

Rule. — Multiply the diameter of the cylinder in inches oy *074. 

Example 1. — Let 40 inches be the diameter of the cylinder. 

Then 40 inches x "074 = 2*96 inches, which is the proper 
depth of the eye round the end studs of the side lever in this en- 
gine. 

Example 2. — Let 64 inches be the diameter of the cylinder. 

Then 64 inches x *074 = 4*736 inches, which is the proper 
depth of the eye round the end studs of the side lever in this en- 
gine. 

It is clear that the diameter of the end stud added to twice 
the depth of the metal running round it will be equal to the 
depth of the side lever at the end 

Hence 2*1 + twice 2*96 = 8*72 will be the depth in inches of 
the side lever at the ends in the engine with the 40-inch cylin- 
der, and 4*48 + twice 4*736 = 13*95 will be the depth in inches 
of the side lever at the ends in the engine with the 64-inch cyl- 
inder. 



DIMENSIONS OF THE CRANK. 271 

TO FLXD THE THICKNESS OF THE EYE EOUND THE END STUDS OF 

SIDE LEYEE. 

Rule. — Multiply the diameter of the cylinder in inches oy *052, 

Example 1. — Let 40 inches be the diameter of the cylinder. 

Then 40 inches x *052 = 2*08 inches, which is the proper 
thickness of eye of side lever round the end studs in this engine. 

Example 2. — Let 64 inches be the diameter of the cylinder. 

Then 64 inches x *052 = 3 '328 inches, which is the proper 
thickness of eye of side lever round the end studs in this engine. 

THE CKANK. 

TO FLND THE PEOPEE DLAUETEE OF THE CBANK-PLN JOUENALS. 

Eule. — Multiply the diameter of the cylinder in inches oy *142. 

Example 1. — Let 40 inches be the diameter of the cylinder. 

Then 40 inches x *142 = 5*680 inches, which is the proper 
diameter of the crank-pin journal in this engine. 

Example 2. — Let 64 inches be the diameter of the cylinder. 

Then 64 inches x *142 = 9*080 inches, which is the proper 
diameter of the crank-pin journal in this engine. 

TO FEND THE PEOPEE LENGTH OF THE CEANE-PIN JOUENAL. 

Eule. — Multiply the diameter of the cylinder in inches dy '16. 

Example 1. — Let 40 inches be the diameter of the cylinder. 

Then 40 inches x "16 = 6 '40 inches, which is the proper 
length of the crank-pin journal in this engine. 

Example 2. — Let 64 inches be the diameter of the cylinder. 

Then 64 inches x *16 = 10*24 inches, which is the proper 
length of the crank-pin journal in this engine. 

TO FLND THE PEOPEE THICKNESS OF THE SMALL EYE OF CBANK. 

Eule. — Multiply the diameter of the cylinder in inches dy *063. 

Example 1. — Let 40 inches be the diameter of the cylinder. 
Then 40 inches x '063 = 2*52 inches, which is the proper 
thickness of the small eye of the crank in this engine. 



272 PROPORTIONS OF STEAM-ENGINES. 

Example 2. — Let 64 inches be the diameter of the cylinder. 
Then 64 inches x '063 = 4*032 inches, which is the proper 
thickness of the small eye of the crank in this engine. 

TO FIND THE PEOPEE BEEADTH OP THE SMALL EYE OF THE CEANK-. 

Rule, — Multiply the diameter of the cylinder in inches by '187. 

Example 1. — Let 40 inches be the diameter of the cylinder. 

Then 40 inches x "187 = 7'48 inches, which is the proper 
breadth of the small eye of the crank in this engine. 

Example 2. — Let 64 inches be the diameter of the cylinder. 

Then 64 inches x "187 = 11*968 inches, which is the proper 
breadth of the small eye of the crank in this engine. 

TO FIND THE PEOPEE THICKNESS OF THE WEB OF CEANK, SUP- 
POSING IT TO BE CONTINUED TO CENTEE OF CEANK PIN. 

Rule. — Multiply the diameter of the cylinder in inches oy *11. 

Example 1. — Let 40 inches be the diameter of the cylinder. 

Then 40 inches x *11 = 4*40 inches, which is the proper 
thickness of the web of crank in this engine, supposing it to be 
continued so far as centre of pin. 

Example 2. — Let 64 inches be the diameter of the cylinder. 

Then 64 inches x *1 1 = 7'04 inches, which is the proper 
thickness of the web of the crank in this engine, supposing that 
the thickness were to be continued to the centre of the crank- 
pin and to be there measured. 

TO FIND THE PEOPEE THICKNESS OF THE WEB OF THE CEANK, 
SUPPOSING THE THICKNESS TO BE CONTINUED TO THE CENTEE 
OF THE PADDLE-SHAFT. 

Rule. — Multiply the square of the length of the crank in inches 
oy 1*561, and then multiply the square of the diameter of 
cylinder in inches oy *1235. Multiply the square root of the 
sum of these products oy the square of the diameter of the cyl- 
inder in inches / divide this quotient oy 360 ; finally extract 
the cube root of the quotient. The result is the thickness of 
the web of the crank at paddle shaft centre in inches. 

Example 1. — What is the proper thickness of the web of crank 



DIMENSIONS OF THE CRANK. 273 

at the centre of the paddle-shaft, supposing the thickness to be 
continued thither and there measured, in the case of an engine 
with a diameter of cylinder of 64 inches and stroke of 8 feet. 

48 = length of crank in inches 
48 



2304 

1-561 constant multiplier 



3596-5 
505-8 product of 64 2 and '1235 



4102-3 



64 = diameter of cylinder 
64 



4096 

•1235 

505-8 

and y 4102-3 = 64 05 nearly 

4096 = square of diameter 



360)262348-5 



128-15 



And y 728 = 9 nearly, which is the proper thickness in inches of the 
crank of this engine measured at the centre of the paddle shaft. 

Example 2. — "What is the proper thickness of the web of crank 
at paddle-shaft centre in the case of an engine with a cylinder 
40 inches in diameter and stroke of 5 feet ? 

30 = length of crank in inches 
30 

900 = square of length of crank 
1*561 = constant multiplier 

1404-9 

12* 



274 PKOPOETIONS OF STEAM-EXGIXE5. 

40 = diameter of cylinder in inches 
40 

1600 = square of diameter of cylinder 
235 = constant multiplier 



197-9 
1404-9 

1602-8 and x/1602-8 = 40-03 
1600 



360)64048 



177-9 
And %J 177'9 = 5-62, which is the proper thickness in inches of the "web 
of the crank, supposing the Treb to be continued to the centre of the 
paddle shaft. 

TO IIXD THE PBOPEE BEEADTH OF THE "WEB OF THE tTRAXTT AT 
PIX-CENTEE, SUPPOSING IT TO BE CONTINUED TO THE CENTEE 
OF THE CBANE-PTN. 

Rule. — Multiply the diameter of the cylinder by '16. The prod- 
uct is the proper breadth of the web of the crank, supposing 
the web to be continued to the plane of the centre of the cranh- 
pin. 

Example 1. — Let the diameter of the cylinder be 40 inches. 
Then 40 inches x *16 = 6*4, which is the proper breadth in 

inches of the web of the crank in the plane of the centre of the 

crank-pin in this engine. 

Example 2. — Let the diameter of the cylinder be 64 inches. 
Then 64 inches x *I6 = 10*24 inches, which is the proper 

breadth of the web of the crank at the crank-pin end in this 

engine. 

TO FIND THE PEOPEE BEEADTH OF THE CEANE: AT PADDEE- 

CENTEE. 

Rule. — Multiply the square of the length of crank in inches by 
1-561, and then multiply the square of the diameter of cyl- 
inder in inches by "1235 / multiply the square root of the 
sum of these products by the square of the diameter of the cyU 



DIMENSIONS OF THE CRANK. 275 

inder in inches; divide the product by 45. Finally \ extract 
the cube root of the quotient. 

Example 1. — What is the proper breadth of the crank at 
paddle-centre in the case of an engine with a diameter of cylin- 
der of 64 inches and stroke of 8 feet ? 

48 length of crank in inches 
48 



2304 

1*561 constant 


multiplier 


3596-5 
505-8 




4102-3 




64 diameter of cylinder 
64 

4086 

•1235 constant multiplier 


505-8 




and ^4102-3 = 64-05 nearly 
4096 


45)262348-5 





5829-97 and ^/5829 = 18 nearly, which is 
the proper breadth in inches of the web of the crank at the shaft-centre 
in this engine. 

Example*}.— "What is the proper breadth of crank at paddle- 
centre in the case of an engine with a diameter of cylinder of 40 
inches and stroke of 5 feet ? 

30 = length of crank in inches 
30 

900= square of length of crank 
1-561 

1404-9 



276 PROPORTIONS OF STEAM-ENGINES. 

40 = diameter of cylinder 
40 



1600 = square of diameter of cylinder 
•1235 



197-6 
1404-9 



1602-5 
and VI 602-5 =40-03 
1600 



45)64048 



1466-7 
and ^1466-7 = 11-24 nearly. 

The purpose of taking the breadth and thickness of the web 
of the crank at the shaft and pin-centres is to obtain fixed points 
for measurement. For, although the web of the crank does not 
extend either to the centre of the shaft or to the centre of the 
pin, it can easily be drawn in as if extending to those points, 
and the breadth and thickness being then laid down at those 
points the proper amount of taper in the web of the crank will 
be obtained. 

TO FIND THE PEOPEE THICXXESS OE THE LAEGE ETE OF THE 

CEANK. 

Rule. — Multiply the square of the length of the cranio in inches 
~by 1'561, then multiply the square of the diameter of the cyl- 
inder in inches ~by *1235 ; multiply the sum of these products 
dy the square of the cylinder in inches; divide the quotient 
hy the length of the crank in inches ; afterwards divide the 
product ~by 1828'28. Finally, extract the cute root of the 
quotient. The result is the proper thickness in inches ofthi 
large eye of crank. 

Example 1. — What is the proper thickness of large eye of the 
the crank in the case of an engine with a diameter of cylinder 
of 64 inches and stroke of 8 feet ? 



DIMENSIONS OF THE CRANK 277 

48 = length of crank in inches 
48 



2304 = square of length of crank 
1*561 = constant multiplier 



3596-5 
505*8 = product of 64* and 1235 



4102-3 



64 = diameter of cylinder 
64 



4096 

•1235 = constant multiplier 



505-8 



4102-3 

4096 = square of diameter 



48)16803020-8 
1828-28)350062-94 
191-47 



and ^191 -4*7 = 5 , '7'7 nearly, whicn is the proper thickness of the large 
eye of the crank in inches. 

Example 2. — "What is the proper thickness of the large eye 
of crank in the case of an engine with a diameter of cylinder of 
40 inches and with a stroke of 5 feet ? 

30 = length of crank in inches 
30 

900 = square of length of crank 
1-561 constant multiplier 

1404-9 



278 PROPORTIONS OF STEAM-ENGINES. 

40 s= diameter of cylinder 
40 



1600 

•12 Go constant multiplier 



197-6 
1404-9 add 



1602-5 

1600 = square of diameter 



1828-28)2564000 



30)1402-41 



46-74 

and y 46-74 = 3*60, which is the proper thickness in inches of the large 
eye of the crank in this engine, ' 



TO ITXD THE PEOPEE DIAMETEE OF THE PADDLE-SHAFT JOURNAL. 

Rule. — Multiply the square of the diameter of the cylinder in 
inches ~by the length of crank in inches; extract the cube root 
of the quotient. Finally, multiply the result ~by *242. The 
final product is the diameter of the paddle-shaft journal in 
inches. 

Example 1. — "WTiat is the proper diameter of the paddle- 
shaft journal in the case of an engine with a diameter of cylin- 
der of 64 inches and stroke of 8 feet ? 

64 = diameter of cylinder in inches 
64 



4096 square of diameter of cylinder 
48 = lenscth of crank in inches 



198608 
and 3/196608 = 58-148, and 58-148 x '242 = 14*07 inches. 

Example 2. — "What is the proper diameter of the paddle- 
shaft journal in the case of an engine with a diameter of cylinder 
of 40 inches and a stroke of 5 feet ? 



DIMENSIONS OF THE PADDLE-SHAFT. 279 

40 = diameter of cylinder 

1600 = square of diameter of cylinder 
30 = length of crank in inches 

48000 and ^48000 = 36*30 

and 36*30 x -242 = 8'19 inches. 

TO FIND THE PEOPER LENGTH OF THE PADDLE-SHAFT JOTJENAL. 

Rule. — Multiply the square of the diameter of the cylinder in 
inches dy the length of the crank in inches ; extract the cube 
root of quotient; multiply the result ~by '303. The product 
is the length of the paddle-shaft journal in inches. 

Example 1. — What is the proper length of the paddle-shaft 
journal in the case of an engine with a diameter of cylinder 64 
inches and stroke 8 feet ? 

64 = diameter of cylinder 
64 



4096 = square of diameter of cylinder 
48 = length of crank in inches 



196608 and 3/196608 = 58-148 
Length of journal = 58-148 x *303 = l?-60 inches. 

Example 2. — What is the proper length of the paddle-shaft 
journal in the case of an engine with a diameter of cylinder of 
40 feet and stroke of 5 feet ? 

40 
40 



1600 
30 



48000 and V 48000=36-30 x -303=10-99. 

It will he seen from these examples that the length of the 
paddle-shaft journals is 1|- times the diameter. The paddle- 
shafts, cranks, and all the other working parts of marine 



£80 PROPORTIONS OF STEAM-ENGINES. 

engines are made of wrought-iron, except the side lever^ 
which are of cast-iron, and the air-pump rod, which is of 
copper or brass. 

THE AIR-PUMP. 

TO FIND THE PEOPEE DIAMETER OF AIR-PUMP. 

Rule. — Multiply the diameter of the cylinder in inches hy *6. 

Example 1. — Let 40 inches be the diameter of the cylinder. 

Then 40 inches x *6 = 24*0 inches, which is the proper diam 
eter of the air-pump in this engine. 

Example 2. — Let 64 inches be the diameter of the cylinder. 

Then 64 inches x *6 = 38*4 inches, which is the proper diam- 
eter of the air-pump in this engine. 

AIR-PUMP ROD. 

TO FIND THE PEOPEE DIAMETER IN INCHES OF THE AIR-PUMP 
ROD WHEN OF COPPER. 

Rule. — Multiply the diameter of the cylinder in inches oy *067. 

Example 1. — Let the diameter of the cylinder be 40 inches. 

Then 40 X *067 = 2*68 inches, which is the proper diameter 
of the air-pump rod when of copper in this engine. 

Example 2. — Let the diameter of the cylinder be 64 inches. 

Then 64 x *067 = 4*28 inches, which is the proper diameter 
of the air-pump rod when of copper in this engine. 

TO FIND THE PROPER DEPTH OF GIBS AND CUTTER THROUGH 
THE AIR-PUMP CROSSHEAD IN INCHES. 

Rule. — Multiply the diameter of the cylinder in inches dy '063, 

Example 1. — Let the diameter of the cylinder be 40 inches. 

Then 40 x "063 = 2'52 inches, which is the proper depth of 
gibs and cutter through the air-pump crosshead in this engine. 

Example 2. — Let the diameter of the cylinder be 64 inches. 

Then 64 x *063 = 4*03 inches, which is the proper depth of 
gibs and cutter through the air-pump crosshead in this engine. 



DIMENSIONS OF PARTS OF THE AIR-PUMP. 281 

TO FIND THE PEOPEE THICKNESS OF GIBS AND GUTTER 
THROUGH AIE-PUMP OEOSSHEAD IN INCHES. 

Rule. — Multiply the diameter of the cylinder in inches by '013. 

Example 1. — Let the diameter of the cylinder be 40 inches. 

Then 40 x '013 = *52 inches, which is the proper thickness 
of gibs and cutter through the air-pump crosshead in this engine. 

Example 2.— Let the diameter of the cylinder be 64 inches. 

Then 64 x *013 = '83 inches, which is the proper thickness 
of gibs and cutter through the air-pump crosshead in this engine. 

TO FIND THE PEOPEE DEPTH LN INCHES OF THE CUTTEE 
THEOUGH THE AIR-PUMP BUCKET. " 

Rule. — Multiply the diameter of the cylinder in inches dy '051. 

Example 1. — Let the diameter of the cylinder be 40 inches. 

Then 40 x '051 = 2*04 inches, which is the proper depth 
of the cutter through the air-pump bucket in this engine. 

Example 2. — Let the diameter of the cylinder be 64 inches. 

Then 64 x '051 = 3*26 inches, which is the proper depth of 
the cutter through the air-pump bucket in this engine. 

TO FIND THE PEOPEE THICKNESS OF THE CUTTEE THEOUGH THE 
AIE-PUMP BUCKET IN INCHES. 

Rule. — Multiply the diameter of the cylinder in inches 7>y *021. 

Example 1. — Let the diameter of the cylinder be 40 inches. 

Then 40 x '021 = "84 inches, which is the proper thickness 
of the cutter through the air-pump bucket in this engine. 

Example 2. — Let the diameter of the cylinder be 64 inches. 

Then 64 x *021 = 1*34 inches, which is the proper thickness 
of the cutter through the air-pump bucket in this engine. 

The cutter through the air-pump bucket should be always 
made of brass or copper, but the gibs and cutter through the air- 
pump crosshead will be of iron. The air-pump bucket should 
always be of brass, and it is advisable to insert the rod into the 
crosshead and also into the bucket with a good deal of taper, so 
as to facilitate its detachment should the bucket require to bo 
taken out. It is usual to form the part of the rod projecting 



BS2 PROPORTIONS OF STEAM-ENGINES. 

throngh the crosshead into a screw, and to screw a nnt upon it, 
This also is a common practice at the top of the piston rod and 
at the bottom of the connecting-rod. 

AIR-POLP CROSSHEAD. 

TO FLND THE FBOPEB DEPTH OF THE EYE OF THE ALB-PUMP 

CEOSSHEAD. 

Rule. — Multiply the diameter of the cylinder in inches oy *1T1. 

Example 1. — Let 40 inches he the diameter of the cylinder. 

Then 40 inches x *171 = 6"84 inches, which is the proper 
depth of eye of air-pnmp crosshead in this engine. 

Example 2. — Let 64 inches he the diameter of the cylinder. 

Then 64 inches x '171 = 10 - 944 inches which is the proper 
depth of the eye of air-pnmp crosshead in this engine. 

TO FIND THE PEOPEE DEPTH OF THE ALB-PUMP CEOSSHEAD AT 
THE MIDDLE OF THE WEB. 

Rule. — Multiply the diameter of the cylinder in inches oy *161. 

Example 1. — Let 40 inches be the diameter of the cylinder. 

Then 40 inches x *161 = 6*44 inches, which is the proper 
depth at the middle of the web of the air-pnmp crosshead in this 
engine. 

Example 2. — Let 64 inches be the diameter of the cylinder. 

Then 64 inches x *161 = 10'30 inches, which is the proper 
depth at the middle of the web of the air-pnmp crosshead in this 
engine. 

TO FIND THE PEOPEE DEPTH OF THE WEB OF THE A IB-PUMP 
CEOSSHEAD AT JOUENALS. 

Rule. — Mutiply the diameter of the cylinder in inches oy *061. 

Example 1. — Let 40 inches be the diameter of the cylinder. 

Then 40 inches x "061 = 2*44 inches, which is the propei 
depth of the web of the air-pnmp crosshead at the jonrnals iD 
this engine. 

Example 2. — Let 64 inches be the diameter of the cylinder. 



DIMENSIONS OF AIR-PUMP CROSSHEAD. 283 

Then 64 inches x •001 = 3*90 inches, which is the proper 
depth of the web of the air-pump crosshead at the journals in 
this engine. 

TO FIND THE PEOPEE THICKNESS OF THE EYE OF THE AIE- 
PTTMP OEOSSHEAD. 

&ULE. — Multiply the diameter of tlie cylinder in inches oy *025. 

Example 1. — Let 40 inches be the diameter of the cylinder. 

Then 40 inches x *025 = 1*00 inches, which is the proper 
thickness of the eye of the air-pump crosshead in this engine. 

Example 2. — Let 64 inches be the diameter of the cylinder. 

Then 64 inches x '025 = 1*600 inches, which is the proper 
thickness of the eye of the air-pump crosshead in this engine. 

TO FIND THE PEOPEE THICKNESS OF THE WEB OF THE AIE-PUMP 
CEOSSHEAD AT THE MIDDLE. 

Rule. — Multiply the diameter of the cylinder in inches ~by *043. 

Example 1. — Let 40 inches be the diameter of the cylinder. 

Then 40 inches x *043 = 1*72 inches, which is the proper 
thickness of the web of the air-pump crosshead at the middle 
in this engine. 

Example 2. — Let 64 inches be the diameter of the cylinder. 

Then 64 inches x *043 = 2*75 inches, which is the proper 
thickness of the web of the air-pump crosshead at the middle 
in this engine. 

TO FIND THE PEOPEE THICKNESS OF THE WEB OF THE AIE-PUMP 
OEOSSHEAD AT THE JOUENALS. 

Rule. — Multiply the diameter of the cylinder in inches oy *037. 

Example 1. — Let 40 inches be the diameter of the cylinder. 

Then 40 inches x *087 = 1*48 inches, which is the proper 
thickness of the web of the air-pump crosshead at the journals 
in this engine. 

Example 2. — Let 64 inches be the diameter of the cylinder. 

Then 64 inches x '037 = 2*36 inches, which is the proper 
thickness of the web of the air-pump crosshead at the journals 
hi this engine. 



284 PllOPORTIONS OF STEAM-ENGINES. 

TO FIND THE PEOPEE DIAMETEE OF THE JOURNALS OF THE AIR- PUMP- 

OEOSSHEAD. 

Rule. — Multiply the diameter of the cylinder in inches oy '051. 

Example 1. — Let 40 inches be the diameter of the cylinder. 

Then 40 inches x *051 = 2*04 inches, which is the proper 
diameter of the journals of the air-pump crosshead in this en- 
gine. 

Example 2. — Let 64 inches be the diameter of the cylinder. 

Then 64 inches x *051 = 3*26 inches, which is the proper 
diameter of the journals of the air-pump crosshead in this en- 
gine. 

TO FIND THE PEOPEE LENGTH OF THE JOUENALS OF THE AIE-PUMP 

OEOSSHEAD. 

Eule. — Multiply the diameter of the cylinder in inches oy *058. 

Example 1. — Let 40 inches be the diameter of the cylinder. 

Then 40 inches x '058 = 2 - 32 inches, which is the proper 
length of the end journals for the air-pump crosshead in this 
engine. 

Example 2. — Let 64 inches be the diameter of the cylinder. 

Then 64 inches x *058 == 3'71 inches, which is the proper 
length of the end journals for the air-pump crosshead in this 
engine. 

AIR-PUMP SIDE RODS. 

TO FIND THE PEOPEE DIAMETEE OF AIE-PUMP SIDE EOD AT THE ENDS. 

Rule. — Midtiply the diameter of the cylinder in inches oy '039. 

Example 1. — Let 40 inches be the diameter of the cylinder. 

Then 40 inches x '039 = 1*56 inches, which is the proper 
diameter of air-pump side rod at ends in this engine. 

Example 2. — Let 64 inches be the diameter of the cylinder. 

Then 64 inches x *039 = 2*49 inches, which is the proper 
diameter of air-pump side rod at ends in this engine. 

TO FIND THE BEEADTH OF BUTT FOE AIE-PUMP SIDE EODS. 

Rule. — Multiply the diameter of the eylinder in inches oy 046i 



DIMENSIONS OF AIR-PUMP SIDE RODS. 285 

Example 1. — Let 40 inches be the diameter of the cylinder. 

Then 40 inches x '040 = 1*84 inches, which is the proper 
breadth of butt for air-pump side rod in this engine. 

Example 2. — Let 64 inches be the diameter of the cylinder. 

Then 64 inches x *046 = 2*94 inches, which is the proper 
breadth of butt of air-pump side rod in this engine. 

TO fi>:d ties peopee thickness of butt FOE AIE-PUMP SIDE EOD. 
Rule. — Multiply the diameter of the cylinder in inches oy '037. 

Example 1. — Let 40 inches be the diameter of the cylinder. 

Then 40 inches x '087 = 1 "48 inches, which is the proper 
thickness of butt for air-pump side rod in this engine. 

Example 2. — Let 64 inches be the diameter of the cylinder. 

Then 64 inches x "037 = 2*36 inches, which is the proper 
thickness of butt for air-pump side rod in this engine. 

TO FIND THE MEAN THICKNESS OF STSAP AT CUTTEE OF AIE-PUMP 

SIDE EOD. 

Rule. — Multiply the diameter of the cylinder in inches oy '019, 

Example 1. — Let 40 inches be the diameter of the cylinder. 

Then 40 inches x '019 = '76 inches, which is the proper 
mean thickness of the strap at cutter of air-pump side rod in 
this engine. 

Example 2. — Let 64 inches be the diameter of the cylinder. 

Then 64 inches x "019 =1*21 inches, which is the proper 
mean thickness of the strap at cutter of air-pump side rod in 
this engine. 

TO FIND THE PEOPEE MEAN THICKNESS OF THE STEAP BELOW GUI 
TEE OF AIE-PUMP SIDE EOD. 

Rule. — Multiply the diameter of the cylinder in inches oy '014. 

Example 1. — Let 40 inches be the diameter of the cylinder. 

Then 40 inches x *014 = *56 inches, which is the proper 
mean thickness of the strap below cutter in the air-pump side 
rod of this engine. 

Example 2. — let 04 inches be the diameter of the cylinder. 



28Q PROPORTIONS OP STEAM-ENGINES. 

Then 64 inches x '014 = *89 inches, -which is the proper 
mean thickness of strap below cntter in the air-pnmp side rod 
of this engine. 

TO FIND THE PEOPEE DEPTH OF THE GIBS AND OUTTEE FOR AIE- 

PUMP SIDE EOD. 

IIule. — Multiply the diameter of the cylinder in inches by "048. 

Example 1. — Let 40 inches be the diameter of the cylinder. 

Then 40 inches x •048 = 1*92 inches, which is the proper 
depth of gibs and cntter for air-pnmp side rod in this engine. 

Example 2. — Let 64 inches be the diameter of the cylinder. 

Then 64 inches x "048 = 3*07 inches, which is the proper 
depth of gibs and cutter for the air-pnmp side rod in this engine. 

TO FIND THE PEOPEE THICKNESS OF THE GIBS AND OUTTEE FOE THE 
ATE-PUilP SIDE EOD. 

Rule. — Divide the diameter of the cylinder in inches ~by 100. 

Example 1. — Let 40 inches be the diameter of the cylinder. 

Then 40 inches -j- 100 = '40 inches, which is the proper 
thickness of the gibs and cutter for the air-pump side rod in this 
engine. 

Example 2. — Let 64 inches be the, diameter of the cylinder. 

Then 64 inches -i- 100 = '64 inches, which is the proper 
thickness of the gibs and cutter of the air-pump side rod in this 
engine. 

It will be satisfactory to compare the dimensions of the parts 
of engines with the actual dimensions obtaining in some engines 
of good proportions which have for some time been in success- 
ful operation ; and I select for the purpose of this comparison 
the side-lever engines constructed by Messrs. Caird & Co., for 
the "West India Mail steamers ' Clyde,' ' Tweed,' ' Tay,' and 
' Tevoit.' The dimensions of the main parts given by the rules, 
and the actual dimensions, are exhibited in the following table, 
touching which it is sufficient to remark that where there is any 
appreciable divergence between the two, the dimensions given 
by the rules appear to be the preferable ones : — 



RULES TESTED BY PRACTICAL EXAMPLES. 



287 



COMPARISON OF DIMENSIONS GIVEN BY THE FOREGOING EULE9 
WITH THE ACTUAL DIMENSIONS OF THE MAIN PAETS OF THE 
SIDE LEYEE ENGINES OF THE STEAMEES ' CLYDE,' ' TWEED,' 

'tay,' and 'teviot,' of 450 hoeses POWEE, CONSTRUCTED 
BY MESSES. CAIRD & CO. 



Diameter of Paddle-Shaft Journal. 


Dimensions 
by Rules. 


Actual 
Dimensions. 


Diameter of paddle-shaft journal 


15-15 

27-84 

10-49 

19-8177 

13-S75 

9 8 

8-14 
12-21 

7-4 

7-03 

9-98 

4-77 

6-6 
13-5 
21-1S3 

6-349 

5-3 
19-85 

4-514 

7-511 

0-0367 x 

P*xD= 

13-579 


15-25 


1 Diameter of crank pin journal 


27-S75 
9-5 ' 




20-625 
13-25 
10-5 
9-75 
15-0 
7-75 
7-6 
9-25 
5-0 
6-375 
14-5 
21-25 
6-375 
5-5 
19-5 
4-875 
9-75 

I 11-5 


Thickness of web of crank at paddle centre 

" w at crank pin centre 

Breadth of crank at crank pin centre 






t " " atmiddle 

" side rod at ends 


" " atmiddle 




tDcpth of web of crosshead at centre 


Depth of web of crosshead at journal 





The rules give generally smaller numbers than Messrs. Caird's 
practice. The difference is greatest in 'Breadth of crank at 
crank-pin centre,' and in ' Exterior diameter of eye of crosshead,' 
and ' Depth of web of crosshead of journals.' 

In five cases above, marked thus f, the rales give greater 
strength than the example selected of Messrs. Caird's engine, 
especially in ' Diameter of main centre,' where Messrs. Caird'3 
proportions are quite too small. 

I have already explained that from any one drawing, all sizes 
of engines of that particular form may be constructed by merely 
altering the scale ; and all the dimensions of ships and engines, 
and, in fact, every quantity whatever which increases or dimin- 
ishes in a given ratio, or according to a uniform law, may be ex- 
pressed graphically by a curve, which will have its correspond- 
ing equation, though sometimes that equation will be too com 
plicated to be numerically expressible. Mr. "Watt, in his earl} 



288 PROPORTIONS OF STEAM-ENGINES. 

practice, laid down most of the dimensions of his engines to 
curves, and, indeed, was in the habit of using that mode of in- 
vestigation and expression in all his researches. A table of the 
dimensions of the parts of engines may easily be laid down in 
the form of a curve ; and the benefit of that practice is, that if 
we have a certain number of points in the curve, we can easily 
find all the intermediate ones by merely measuring with a pair 
of compasses and a scale of equal parts. Thus, for example, we 
may lay down the table of the diameter of crank-shaft, given in 
page 294, to a curve as follows : — First draw a straight horizon- 
tal line, which divide into equal parts by any convenient scale, 
beginning, as in the table, with 20, and ending with 100. If now 
we erect vertical lines or ordinates at every division of the hori- 
zontal line, and if, with any given length of stroke, say 2 feet, 
we know the diameter of shaft proper for some of the diameters 
of cylinder — say for a 20-inch cylinder, 4*08 inches ; for a 24-inch 
cylinder, 4'66 inches; for a 40-inch cylinder, 6*55 inches; and 
for an 80-inch cylinder, 10*29 inches — we can easily determine 
the diameters of shaft proper for all the intermediate diameters 
of cylinders, by marking off with the same scale, or any other, 
the vertical heights corresponding to all the diameters we know; 
and a curve traced through these points will intersect all the 
other ordinates, and give the diameters proper for the whole 
series. By thus setting down the known quantities in order to 
deduce the unknown, we shall at the same time see whether the 
quantities we set down follow a regular law of increase or not ; 
for if they do not, instead of all the points marked off falling 
into a regular curve, some of them will be above the curve and 
some of them beneath it, thus showing that the quantities given 
do not form portions of a homogeneous system. If the quantity 
increases in arithmetical progression, the curve will become a 
straight angular hue. Thus in the case of the diameter of the 
piston rod, as the increase follows the same law as the increase 
of the diameter of the cylinder, the law of increase will be ex- 
pressed by a right-angled triangle, the diameters of the cylinder 
being represented by the divisions on the base, and the diameters 
of piston rod by the corresponding vertical ordinates. If to the 



RULES TESTED BY PRACTICAL EXAMPLES. 289 

curve of diameter of crank shaft for each diameter of cylinder 
with any given length of stroke, we add below the base another 
curve pointing downwards, representing the increase of the di- 
ameter of the shaft due to every increase of the length of the 
stroke, the diameter of the cylinder remaining the same, the total 
height of the conjoint ordinates will show the diameter of the 
shaft for each successive diameter of cylinder and length of 
stroke. One of the curves will be convex and the other con- 
cave, and the convexity of the one will be equal to the concavity 
of the other, so that the ordinates will be the same as those of a 
triangle. Hence, if we double the diameter of the cylinder, and 
also double the length of the stroke, we shall double the diam- 
eter of the shaft ; if we treble the diameter of the cylinder, and 
also treble the length of the stroke, we shall treble the diameter 
of the shaft, and so on in all other proportions. By referring to 
the table in page 294, we shall see that these relations are there- 
preserved. Thus a 20-inch cylinder and a 2-feet stroke has a 
shaft of 4'08 inches in diameter ; a 40-inch cylinder and a 4-feet 
stroke, a shaft of 8*16 inches diameter; a 60-inch cylinder and 
a 6-feet stroke, a shaft 12*25 inches diameter, and so on. If this 
were not so, an engine drawn on any one scale would not be ap- 
plicable to any other of a different size ; whereas we know that 
any one drawing will do for all sizes of engines by merely chang- 
ing the scale. 

It is very convenient in making drawings of engines to adopt 
some uniform size for the drawing-boards and drawings, and to 
adhere to them on all occasions. The best arranged drawing- 
office I have met with is that of Boulton and "Watt, which was 
originally settled in its present form by Mr. "Watt himself, who 
brought the same good sense and habits of methodical arrange- 
ment to this problem that he did to every other. The basis of 
Boulton and "Watt's sizes of drawings is the dimensions of a sheet 
of double elephant drawing paper; and all their drawings are 
either of that size, of half that size, or of a quarter that size, 
leaving a proper width for margin. The drawing-boards are all 
made with a frame fitting around them, so that it is not neces? 
sary to glue the paper round the edges; but the damped sheet 
13 



290 



PROPORTIONS OF STEAM-ENGINES. 



TABLE OF THE DIMENSIONS OF THE PBLNC1PAL 

MAPJXE EXGIXES OF 



NAMES OF PAETS. 



Diameter of 

Cylinder 

Piston rod 

Air-pump 

Air-pnmp rod. 

Injection cock 

Hot-water pump 

Feed-pipe 

Steam-pipe 

Waste-water pipe 

Beam gudgeon , 

Pins in beam ends 

Air-pump pins in beam 

Crank-pin , 

Main shaft 

Paddle-wheels, in feet 

Weight-shaft bearings , 

Stroke of 

Piston , 

Air-pump bucket 

Feed-pump plunger 

Cylinder crosshead 

Depth of boss , 

Diameter of boss 

Depth of middle 

Thickness , 

Air-pump crosshead 

Depth of boss 

Diameter of boss 

Depth of middle 

Thickness 

Columns 

Diameter at top 

Diameter at bottom 

Centre to centre of 

Air-pump, side-rods transversely 

Beams w u 

Frames * " 

Engines " " 

Length of steam port 

Breadth of steam port 

Fort valve passage 

Depth 

Width 

Beam. 

Breadth at middle 

Breadth at ends 

Thickness 



in. 


in. 


m. 


20 


24 


27 


2 


22 


2} 


1-2 


15 


17 


H 


H 


2 


1} 


1* 


!-§■ 


2} 


u 


3 


H 


1} 


2 


4 


o 


5} 


5 


6 


7 


Si 


4} 


o 


2 


2-| 


2* 


H 


If 


H 


2} 


o 


3} 


4} 


5} 


6} 


9 


11 


11 


2 


2i 


21 


24 


30 


30 


12 


15 


15 


6 


n 


7} 


6 


n 


8 


4 


4} 


5 





5} 


61 


If 


U 


H 


4} 


5 


5* 


n 


H 


31 


3} 


4 


4+ 


1 


n 


1} 


4 


4f 


5i 


4} 


5} 


«f 


29} 


34} 


37} 


33 


39 


42} 


21 


23 


25J 


66 


72 


76 


n 


8} 


10 


H 


If 


2 


2 


2 


2} 


13 


14 


15} 


14 


18 


19 


5 




61 


1 


n 


14 



EXAMPLES OF APPROVED DIMENSIONS. 



291 



PAET3 OF MESSES. MATIDSLAY, SONS, AXD FIELD S 
DIFFEEEXT POTYEES. 



POATEE 


OF EXGLNE. 
















Pi 


Pi 


pi 


Pi 


Ph' 


pi 


Pi 


Pi 


Pi 


Pi 


s 


W 


h 




hrt 


R 


H" 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


1H 


C3 


CO 


-* 


ira 


CO 


k- 


CO 


o 


T-t 


1-1 


rH 


in. 


in. 


in. 


in. 


in. 


in. 


in. 


in. 


in. 


in. 


82 


36} 


40 


43 


46 


48 


50 


52} 


55} 


57 


3} 


s* 


4 


4} 


4} 


4} 


4£ 


5 


5} 


5} 


is} 


21 


23 


24 


26 


27} 


28 


80 


31} 


34 


2} 


2i 


2} 


2£ 


3 


3} 


3} 


3} 


4 


4} 


2 


2i 


2i 


2} 


3 


3£ 


3} 


3} 


3*. 


3} 


3* 


4 


4} 


4} 


5 


5} 


6 


6} 


7 


7} 


2} 


2} 


2* 


2f 


3 


3} 


34- 


3} 


3} 


4 


6* 


7 


7f 


8} 


9} 


10 


10} 


11 


11} 


12 


8 


9 


9* 


10 


10} 


11} 


12} 


13 


13} 


14 


5* 


6 


6} 


7 


7} 


8 


s* 


9 


9} 


9} 


Si 


3} 


4 


4} 


4} 


4} 


5 


5} 


5} 


5} 


2 


2i 


2} 


2} 


2} 


2} 


3 


3£ 


%k 


3} 


4 


4} 


5 


5* 


6 


6} 


7 


T# 


7} 


8 


7 


n 


8} 


9i 


10 


10} 


10} 


11} 


12 


12} 


13 


13 


15 


17 


17 


19 


19 


21 


21 


23 


n 


2i 


2} 


2} 


3 


8} 


3} 


3} 


8} 


S| 


86 


86 


42 


48 


52 


56 


60 


63 


66 


72 


38- 


IS 


21 


24 


26 


28 


30 


31} 


33 


36 


9 


9 


10} 


12 


13 


14 


15 


16 


16} 


18 


9} 


10i 


12 


13 


14 


14} 


15 


16 


17 


17} 


6 


6| 


7} 


8 


S} 


9 


9} 


10 


11 


12 


7} 


8} 


9} 


10! 


11} 


11} 


12} 


13 


13} 


14 


2} 


2i 


2f 


3 


3} 


3} 


3} 


3} 


4 


4} 


6i 


8 


9 


10 


10} 


10} 


11 


1H 


12 


12} 


4? 


4* 


5| 


5i 


5} 


6 


6} 


6f 


7} 


7} 


5} 


6i 


7' 


T* 


8 


8} 


8} 


9 


9} 


9} 


1* 


1* 


2 


2} 


2} 


2} 


«f 


2} 


2} 


2} 


6 


7 


8 


8i 


9i 


9} 


9} 


10 


10} 


10} 


6| 


7f 


9 


9£ 


10} 


10} 


11 


11} 


11} 


12 


42} 


47} 


53 


55} 


60} 


63 


67 


68} 


70 


72 


48 


54 


60 


63 


69 


69 


72 


78 


80 


88 


27 


30 


84 


34 


40 


40 


42 


44 


45 


46 


S4 


88 


96 


100 


108 


108 


112 


126 


128 


130 


1H 


13 


15 


IS} 


18} 


19 


19 


20 


20 


21 


2} 


2* 


3 


3 


4 


4 


4} 


4} 


4} 


4} 


3} 


3} 


4 


4} 


5 


5i 


5} 


6 


0} 


7 


18 


20 


24 


26 


28 


28 


29 


31 


31 


32 


23 


25 


28 


29 


33 


34 


85 


36 


38 


39 


8 


8| 


10 


10} 


12 


12} 


12} 


12 


15 


15* 


1* 


li 


It 


2 


2} 


2f 


2} 


2} 


2} 


2f 



292 



PROPORTIONS OF STEAM-ENGINES. 



TABLE OF THE DIMENSIONS OF THE PEINCIPAli 

ENGINES OF DIF- 



"a 
.9 

a .• 

O CQ 




l . 

fc!g 


J* 

O 


•3 

•a 
a 

<** *» 


.9 
'3 

a 

u a 


o 

■S-S 


a 

o 

'5c 


i ^ 

o ° 

~1 


« o 

2-°" 

■2 *■ 


S 

3 

o 5, 


% 


8-a 

ft 


aa 

aj 3 

5* 


Cfi 


a 

ft 


a 

s 


ft 


a 
3 


a «< 


la* 

.2 3 
ft * 


<0 

a 

(3 
ft 








ft. in. 


ft. 


in. 


in. 


in. 


in. 


in. 


in. 


10 


20 


12 


2 


9 


4 


7 


2 


1* 


1 


4 


15 


24 


15 


2 6 


11 


4f 


8* 


2* 


1* 


1* 


5 


20 


27 


17 


2 6 


11 


5| 


10 


3 


If 


1* 


5f 


30 


31J 


18* 


3 


13 


6* 


lOf 


s* 


If 


1| 


6* 


40 


36* 


20 


3 


13 


7* 


11* 


3* 


2* 


1^ 

-*-8 


7 


50 


39* 


22 


3 6 


15 


8> 


12* 


3| 


2* 


2 


Ti 


60 


43 


24 


4 


17 


9 


13 


4 


2f 


2* 


8* 


70 


4G 


25* 


4 3 


17 


3* 


13f 


4* 


3 


2* 


9* 


80 


48 


27 


4 6 


19 


10 


14* 


4* 


3* 


Sf 


10 


90 


50 


28 


4 9 


19 


10* 


15* 


4| 


3* 


3 


10* 


100 


52* 




5 


21 


11 


16 


5 


3f 


31 


11 


110 


55 




5 


21 


Hi 


16f 


5} 


4 


3* 


11* 


120 


57* 




5 6 


23 


12* 


17* 


5* 


4* 


3* 


12 



is laid upon the board, which it somewhat overlaps, and the 
frame then comes down and turns over the edges of the paper 
upon the sides of the hoard, and the frame being then fixed sc 
that its face is flush with the paper, the paper by being thus 
bound all round the edges is properly stretched when dry. In 
Mr. "Watt's time the drawings were made with copying ink, and 
an impression was taken from them by passing them through a 
roller press, so as to retain the original in the office, while a 
duplicate of it was sent out with the work ; and the copying 
press was invented by Mr. Watt for this purpose. The whole 
of the drawings pertaining to each particular engine are placed 
in a small paper portfolio by themselves ; and these small port- 
folios are numbered and arranged in drawers, with a catalogue 
to tell the particular engine delineated in the drawings of each 
portfolio. In this way I have found that the drawings illustra- 
tive of any engine, though it may have been made in the last 



EXAMPLES OF APPROVED DIMENSIONS. 



293 



PARTS OF MESSES. SEAWAED AND CO.'s MAEINE 
FEEENT POWEES. 



"3 

"5. j3 


cj . 




1 


13 


a 

o 
to . 


a 


to 


a . 

.3 to 





.9 

a. 


9g 


o 2< 




o 


"S 3 


ii 


El 


II 

.C o 


"3.9 
<tf of 

1^ 


M 


-a 

3 




S 


fi 

ft 


to 
a 


c 3- 

oi 

pa 


1 


2"° 


£ to 

% 




0) 

E 
ft 


t3> 

in. 




in. 


in. 


ft. in. 


in. 


in. 


in. 


in. 


in. 


in. 


25 


5 


u 


6 


2 


3 


11 


2* 


2 


21 


2* | 


23 


6 


n 


7 


24 


4 


n 


24 


21 


3 


Si i 


30 


T 


o 


8 


2* 


5 


if 


3 


2* 


Si 


3* j 


35 


8 


21 


8 8 


2-5- 


S£ 


2 


si 


21 


4 


44 


3S 


9 


24 


10 


3 


G 


9i 

-4 


Si 


2J 


4* 


4f 


40 


9i 


2f 


10 G 


3i 


<4 


2* 


3| 


2f 


5 


54 


44 


10 


3 


11 6 


Si 


7 


2f 


4 


2f 


5i 


6 


50 


10 


<^4 


12 6 


3£ 


71 


3 


44 


3 


6 


6* 


55 


11 


si 


13 


4 


8 


3i 

"4 


41 


31 


6i 


7 


58 


12* 


3| 


13 G 


41 


pi 


Si 


4f 


Si- 


7 


74 


62 


13 


4 


1G 


41 


9 


8| 


5 


Si 


7f 


7f 


CO 


13| 


41 


1G 


4| 


91 


3f 


54 


3f 


7| 


81 


70 


14 


H 


17 G 


5 


9f 


31 


5£ 


H 


8 


Si 



century, could be produced to me iu a few minutes ; and the 
system is altogether more perfect and more convenient than any 
other with which I am acquainted. The portfolios are not large, 
which would make them inconvenient, but are of such size that 
a double elephant sheet has to be folded to go into one of them ; 
but most of the drawings are on small sheets of paper, which is 
a much more convenient practice than that of drawing the de- 
tails upon large sheets. 

It will be interesting to compare with the results given in 
the foregoing rules the actual sizes of some side lever engines of 
approved construction. Accordingly I have recapitulated, in the 
tables introduced above, the principal dimensions of the marine 
engines of Messrs. Maudslay and Messrs. Seaward. These tables 
are so clear, that they do not require further explanation, and 
the sjmie remark is applicable to the tables which follow. 



294 



PROPORTIONS OF STEAM-ENGINES. 



DIAMETEPw OF WEOTTGHT-LEOX OEAXK-SHAFT JOCENAL. 



"8.3 


















. 


a u ™ 

■?,-! — 
20 






LENGTH OP STROKE IN FEET. 








2 | 2V 


3 




4 


4i 

5-34 


5 



5-53 


5. 1 


6 


1 


8 


9 
6-73 


4-03 


4-39 


4-67 


4-91 


5-14 


5-72 


5-88 


6-19 


6-4S 


21 


4"23 


4-54 


4-82 


5-05 


5-30 


5-50 


5-71 


5-39 


6-07 


6-39 


6-63 


6-95 


22 


4*37 


4-63 


4-96 


5-20 


5-46 


5-66 


5-83 


6-07 


6-25 


6-53 


6-83 


7-16 


23 


4-52 


4-81 


5-11 


5-35 


5-62 


5-S3 


6-05 


6-25 


6-43 


6-77 


7-08 


7-37 


24 


4-66 


4-95 


5-25 


5-51 


5-73 


5-99 


6-22 


643 


6-62 


6-96 


7-23 


7-53 


25 


4-31 


5-09 


5-40 


5-66 


5-94 


6-16 


6-40 


6-60 


6-SO 


7-15 


7/49 


7-73 


26 


4-95 


5-22 


5-54 


5-S1 


6-10 


6-33 


6-57 


6-78 


6-98 


7-35 


7-69 


7-99 


27 


510 


5-36 


5-69 


5-96 


6-26 


6-49 


6-74 


6-96 


7-16 


7*54 


7-S9 


8-20 


23 


5-24 


549 


5-93 


6-11 


6-42 


6-66 


6-91 


7-14 


7-35 


7-73 


8-09 


8-41 


29 


5-33 


5-53 


5-98 


6-26 


653 


6-32 


7-08 


7-31 


7-53 


7-92 


8-29 


8-62 


30 


5-42 


5-67 


6-12 


6-42 


6-74 


6-99 


7-26 


7-49 


7-72 


8-12 


8-49 


8-83 


31 


5-47 


5-83 


6-25 


6-56 


6-34 


7-14 


7-41 


7'65 


7-83 


S-29 


8-67 


9-02 


32 


5-59 


6-00 


6-38 


6-69 


6-93 


7-29 


7'56 


7-81 


8-04 


S-46 


8-85 


9"21 


33 


5-71 


6-12 


6-51 


6-83 


7-12 


7-43 


7-71 


7-97 


8-20 


8-63 


9-03 


9-39 


34 


5-83 


6-24 


6-64 


6-96 


7"26 


7"53 


7-S7 


813 


8-37 


8'30 


9-21 


9-53 


35 


595 


6-37 


6-77 


7-10 


7-41 


7-73 


802 


8-23 


8-53 


S-97 


9-39 


9-76 


36 


6-07 


6-49 


6-90 


7-23 


7-55 


7-S3 


8-17 


8-44 


8-69 


9-14 


9-57 


9-95 


3T 


6-19 


6-61 


7-03 


7-37 


7-69 


8-03 


8-32 


8-60 


8-S5 


9-31 


9-75 


1044 


33 


6-31 


6-73 


7-16 


7-50 


7-83 


8-17 


S-47 


8-76 


9-02 


9-43 


9-93 


10-33 


39 


6-43 


6-85 


7-29 


7-64 


7-97 


8-32 


8-63 


8-91 


9-18 


9-65 


10-11 


10-51 


40 


6-55 


6-93 


7-42 


7-73 


8-16 


8-47 


S-79 


9-07 


9-3.5 


9-53 


10-29 


10-70 


41 


6-57 


7-19 


7-54 


7-90 


8-29 


8-30 


8-93 


9-21 


9-50 


9-99 


10-45 


10-37 


42 


6-67 


7-20 


7-66 


8-03 


8-42 


8-74 


9-07 


9-36 


9-65 


10-15 


10-62 


11-04 


43 


6-77 


731 


7-73 


8-15 


8-55 


8-87 


9-21 


9-50 


9-30 


10-31 


10-73 


11-21 


44 


6-37 


7-42 


7-90 


8-23 


8-63 


9-01 


9-35 


9-65 


9-95 


10-47 


10-95 


11-38 


45 


6-97 


7-54 


8-02 


8-40 


8-81 


9-14 


9-50 


9-79 


10-10 


10 63 


11-11 


11-55 


46 


7-06 


7-65 


8-15 


8-53 


8-94 


9-23 


9-64 


9-94 


10-25 


10-78 


11-23 


11-72 


48 


7-26 


7-33 


8-40 


8-73 


9-20 


9-55 


9-92 


10-23 


10-54 


11-09 


11-61 


12-08 


50 


7-43 


8-10 


8-61 


9-02 


9-46 


9-82 


10-20 


10-51 


10-34 


11-42 


11-94 


12-41 


52 


7-70 


8-31 


8-33 


9-26 


9-71 


1003 


10-47 


10-79 


11-12 


11-71 


12-25 


12-73 


54 


7-90 


S-52 


9-05 


9-50 


9-96 


10-34 


10-74 


11-05 


11-40 


12-00 


12-56 


13-05 


56 


8-09 


8-73 


9-27 


9-73 


10-21 


10-60 


11-01 


11-33 


11-69 


12-29 


12-37 


13-37 


53 


S-29 


8-94 


950 


9-97 


10-45 


1086 


11-23 


11-61 


11-97 


12-53 


13-13 


13-69 


GO 


S-49 


9-15 


9-72 


10-20 


10-70 


11-12 


11-55 


11-89 


12-25 


12-89 


13-43 


14-02 


62 


8-67 


9-35 


9-93 


10-40 


10 89 


11-34 


11-79 


12-14 


12-51 


13-17 


13-77 


14-32 


64 


8-86 


9-55 


10-14 


10-00 


11-03 


11-56 


12-03 


12-39 


12-78 


13-45 


14-06 


14-62 


68 


9-04 


9-74 


10-35 


10-79 


11-27 


11-73 


12-23 


12-64 


13-04 


13-73 


14-35 


14-93 


63 


9-22 


9-94 


10 56 


10-99 


11-45 


11-99 


1252 


12-39 


13-30 


14-01 


14-64 


15-23 


70 


9-41 


10-14 


10-77 


11-19 


11-64 


12-20 


12-77 


13-16 


13-57 


14-29 


14-94 


15-54 


72 


9-59 


10-33 


10-97 


11-41 


11-90 


12-44 


1301 


13-40 


13-82 


14-55 


15-22 


15-83 


74 


9-76 


10-52 


11-17 


11-62 


12-16 


1267 


13-25 


13-64 


14-07 


14-S2 


15-50 


1612 


7G 


9-93 


10-71 


11-36 


11-33 


12-43 


1291 


13-49 


!3->9 


14-32 


15-09 


15-73 


16-41 


73 


10-11 


10-S9 


11-56 


1205 


1269 


13-15 


13-73 


14-13 


14-57 


15-35 


16-06 


1670 


80 


10-29 


11-03 


11-76 


12-27 


12-96 


13-38 


13-96 


14-33 


14-34 


15-62 


16-33 


16-98 


82 


1046 


11-26 


11-96 


12-49 


13-17 


13-61 


14-19 


14-02 


15-03 


15-83 


16-59 


17-26 ! 


84 


10-63 


11-44 


12-15 


12-71 


13-33 


13-34 


14-42 


14-35 


15-32 


16-13 


16-36 


17-54 


86 


10-30 


11-61 


12-35 


12-92 


13-59 


14-07 


14-65 


15-09 


15.56 


16-38 


17-13 


17-82 


83 


10-97 


11-79 


12-54 


13-14 


13-30 


14-30 


14-33 


15-32 


15-80 


16-63 


17-40 


lylu 


90 


11-13 


11-99 


12-74 


13-34 


14-00 


14-54 


1510 


15-56 


16-05 


16-39 


17-66 


18-37 i 


92 


11-29 


12-17 


12-92 


13-47 


14-21 


14-75 


15-32 


15-79 


16-23 


17-14 


17-92 


13-64 


94 


11*43 


12-34 


13-10 


13-60 


1442 


14-96 


15-54 


16-02 


16-51 


17-33 


1313 


18-91 | 


96 


11-61 


12-51 


13-29 


13-73 


14-63 


15-13 


15-76 


16-25 


16-74 


17-63 


18-44 


19-18 ! 


93 


1177 


12-63 


13-47 


13-33 


14-34 


15-39 


15-93 


16-43 


16-97 


1787 


1S-70 


19-45 


100 


11-93 


12-33 


13-66 


14-01 


15-04 


15-81 


16-20 


1671 


17-22 


13-12 


13-95 


19-71 | 

1 



TABLES OF NOMINAL TOWERS OF ENGINE. 



295 



..xEXGTII OF TVROUGHT-IKON CRANK-SHAFT JOURNAL. 



Diameter of 
Cylinder in 1 
inches. 








LENGTH 


OF STROKE IN FEET. 


2 


H 


3 


H 


4 


*H 


5 
6-93 


H 


6 


1 


8 
8-11 


9 


20 


5-10 


5-49 


5-S4 


6-15 


6-43 


6-69 


7-16 


7-36 


7-75 


8'43 


21 


5-37 


5-69 


6-03 


6-31 


6-62 


6-S7 


7-14 


7-36 


7-59 


7-99 


8-36 


8-69 


22 


5-53 


5-S5 


6-21 


6-50 


6-82 


7-08 


7-35 


7-59 


7-82 


8-23 


8-61 


8-86 


23 


5-68 


6-02 


639 


6-69 


7-02 


7-29 


7-57 


7-S1 


8-05 


847 


8-86 


9-23 


24 


5-84 


6-19 


6-57 


6-88 


7-22 


7-50 


7-78 


804 


8-28 


8-71 


9-11 


9-50 


25 


5-99 


6-36 


6-75 


7-07 


742 


7-70 


8-00 


8-26 


8 51 


8-95 


9-36 


9-76 


26 


6-15 


6-53 


6-93 


7-26 


7-62 


7-91 


8 21 


848 


8-74 


9-19 


9-61 


10-02 


27 


6-30 


6-70 


7-11 


745 


7-82 


8-11 


8-43 


8-70 


8-97 


9-43 


9-86 


10-28 


28 


6-46 


6-S7 


7-29 


7-64 


8-02 


8-32 


8-64 


S-92 


9-20 


9-67 


10-11 


10-54 


29 


6-61 


7-04 


747 


7-S3 


8-22 


8-53 


8-S6 


944 


943 


9-91 


10-36 


10-80 


30 


6-77 


7-21 


7-65 


8-02 


842 


8-74 


9-07 


9-36 


965 


10-15 


10-61 


11-04 


31 


6-92 


7-37 


7-82 


8-19 


8-60 


8-93 


9-27 


9-56 


9*86 


10-37 


10-84 


11-28 


32 


7-06 


7-52 


7-98 


8.36 


8-78 


941 


947 


9-76 


10-06 


10-58 


11-07 


11-52 


33 


7-20 


7-67 


814 


8-53 


8-96 


9-30 


9-56 


9-96 


10-27 


10-79 


11-30 


11-76 


34 


7-34 


7-S2 


8-30 


8-70 


9-14 


94S 


9-75 


10-15 


1047 


11-00 


11-53 


11*99 


35 


7-48 


7-97 


846 


8-87 


9-31 


9-67 


9-94 


10-35 


10-68 


11-22 


11-76 


12-22 


36 


7-62 


8-12 


S-63 


9-04 


949 


9-S5 


1013 


10-55 


10-88 


13-43 


11-98 


12-45 


37 


7-76 


8-27 


8-79 


9-21 


9-67 


10-04 


1032 


10-74 


11-09 


11-64 


12-20 


12-68 


38 


7-90 


8-42 


8 95 


9-38 


9-85 


10-22 


10 51 


10-94 


11-29 


11-S6 


1242 


12-91 


89 


8-05 


8-57 


9-11 


9-55 


10-03 


1041 


10-S0 


11-14 


1149 


12-07 


12-64 


13-14 


40 


8-19 


8-72 


9-27 


9-72 


10-20 


10-58 


10-99 


H-34 


11-69 


12-29 


12-86 


13-37 


41 


8-32 


8-86 


942 


9-S8 


10-37 


10-75 


11-17 


11-53 


11-88 


1249 


13-07 


13-59 


42 


8-44 


9-00 


9-57 


10-03 


10-54 


10-92 


11-34 


11-71 


12 06 


12-69 


13-28 


13-81 


43 


8-56 


9-13 


9-72 


10-18 


10-70 


U-09 


11-52 


H-89 


1225 


12-S8 


1349 


14-03 


44 


8-68 


9-27 


9-87 


10-37 


10-86 


11-26 


11-69 


12-07 


12-43 


13-08 


13-70 


14-25 


45 


8-S0 


940 


10-02 


10 49 


11-02 


H-43 


11-87 


12-25 


12-62 


33-28 


13-91 


1446 


46 


8-92 


9-54 


10-17 


10-64 


11-18 


11-59 


12-04 


1243 


12-80 


1347 


14-12 


14-67 


48 


9-16 


9-81 


1047 


10-94 


11-50 


11-93 


12-39 


12-79 


13-17 


13-87 


14-52 


15-09 


50 


9-40 


10-08 


10-76 


11-26 


11-82 


12-27 


12-75 


13-15 


13-55 


14-27 


14 92 


15-51 


52 


9-65 


10-34 


1104 


11-56 


1213 


12-59 


13-09 


13-50 


13 90 


14-64 


15-32 


15-92 


54 


9-89 


10-60 


11-32 


11-86 


12-44 


12-91 


13-43 


13-85 


14-25 


15-02 


15-71 


16-32 


56 


10-13 


10-86 


11-60 


12-16 


12-75 


13-23 


13-77 


14-19 


14-60 


15-39 


16-09 


16-72 


58 


10-37 


11-13 


11-8S 


1246 


13-06 


13-55 


1411 


14-53 


14-95 


15-76 


1647 


17*12 


60 


10-61 


11-37 


12-15 


12-75 


13-37 


13-87 


14-44 


14-86 


1531 


16-11 


16-85 


17-52 


62 


10-84 


11-62 


1241 


13-00 


13-61 


14-15 


14-76 


15-18 


15-64 


1647 


17-23 


17-90 


64 


11-07 


11-86 


12-67 


13-25 


1385 


1442 


15-06 


15-50 


15-97 


16-83 


17-59 


18 23 


66 


11-30 


12-11 


12-93 


13-50 


1409 


14-70 


15-36 


15-82 


16-30 


17-18 


17-95 


18-66 


68 


11-53 


12-35 


13-19 


13-75 


14-33 


14-97 


15-66 


16-14 


16-63 


17-52 


18-31 


19-04 


70 


11-76 


12-60 


1346 


14-01 


14-55 


15-25 


1596 


1646 


16-96 


17-86 


18-67 


1942 


72 


11-98 


12-S4 


13-71 


14-29 


14-88 


15-56 


16-26 


16-77 


17-28 


18-20 


19-03 


19-7S 


74 


12-20 


13-07 


13-96 


14-5S 


1521 


1587 


16-56 


17-08 


17-60 


18-54 


19-39 


20-14 


76 


12-42 


13-30 


14-20 


14-87 


15-54 


16-18 


16-86 


17-39 


17-92 


18-87 


19-73 


20-50 


78 


12-64 


13-53 


14-44 


15-15 


15-87 


1649 


1716 


17-70 


18-24 


19-19 


20-07 


20-86 


80 


12-S6 


13-76 


14-70 


15-44 


16-20 


16-81 


17-45 


18-00 


18-55 


19-52 


2041 


21-22 


82 


13-07 


13-99 


14-95 


15-70 


1646 


17-08 


17-73 


18-29 


18-86 


19-84 


20-75 


21-58 


84 


13-28 


14-22 


15-19 


15-95 


16-72 


17-35 


1S-03 


18-58 


1916 


20-16 


21-09 


21-84 


86 


33-49 


1445 


15-43 


16-20 


1698 


17-62 


1831 


1S-87 


1946 


20-48 


21-43 


22-28 


88 


13-70 


14-68 


15-67 


1645 


17-24 


17-89 


18*59 


19.16 


19-75 


20-S0 


21-75 


22-62 


90 


33-91 


14-91 


15-92 


16-76 


17-50 


1845 


1S-S7 


1947 


20-06 


21-11 


22-07 


22-96 


92 


1411 


15-14 


16-15 


16-95 


17-76 


1842 


19-14 


19-75 


20-36 


2142 


22-40 


23-30 


94 


14-31 


15-36 


16-38 


17-19 


18-02 


18-69 


1940 


20-01 


20-65 


21-73 


22-73 


23-64 


96 


14-51 


15-59 


16-61 


17*43 


18-28 


18-96 


19-67 


20-33 


20-94 


22-03 


23 05 


23-98 


98 


14-71 


15-82 


16-84 


17-67 


18-54 


19-23 


19'95 


20-57 


21-23 


22 33 


23-37 


24-32 


100 

i 


14-91 


16-08 


17-07 


17-92 


18-80 


19-52 


20-25 


20-S3 


21-52 


22-65 


23-69 


24-64 



296 



PROPORTIONS OF STEAM-ENGINES. 



BEEADTH OE WEB OF CEANK, SUPPOSING IT TO BE CONTINUED 
TO PADDLE-SHAFT CENTRE. 



o.2 
<s u ™ 

20 


LENGTH OF STROKE IN FEET. 


2 

5-28 


2* 


3 


8* 


4 

6-46 


6*72 


5 


*ii 


6 


7 


8 


9 


5-45 


5-62 


6*04 


6-98 


7-19 


7*40 


7-78 


8*12 


8-44 


21 


5-51 


5-68 


5-85 


6-27 


6-68 


6-94 


7-21 


7-42 


763 


8*04 


8*39 


8*71 


22 


5-74 


5-91 


6-08 


6-49 


6-90 


7-16 


7-44 


7-65 


7*S6 


S-30 


8*65 


8-98 


23 


5-96 


6-14 


6-31 


6-72 


7-12 


7-38 


7-67 


7*88 


8-09 


8*55 


8*90 


9*25 


24 


6-18 


6-36 


6-54 


6-94 


7-33 


7-60 


7-89 


811 


8*32 


8*S0 


9*16 


9*52 


125 


6-40 


6-58 


6-77 


7-16 


7*54 


7*S2 


8-11 


8-34 


8-55 


9*04 


941 


9*79 


2.) 


6-62 


6-80 


7-00 


7-38 


7-75 


8-04 


8-33 


8-57 


8*78 


9-28 


967 


10-06 


27 


6-84 


7-03 


7-23 


7-60 


7-96 


8-26 


8-55 


8-80 


9*01 


9-51 


992 


10-33 


28 


7-06 


7-25 


7-46 


7-82 


8-17 


8-48 


8-77 


9*03 


9-24 


9-74 


10*18 


10-59 


29 


7-28 


7-48 


7-69 


8-04 


8-38 


8-70 


8-99 


9-26 


9-47 


9-97 


10-43 


10-85 


80 


7-50 


7-71 


7-92 


8-26 


8-60 


8-92 


9-24 


949 


9*74 


10*22 


10-68 


11*10 


31 


7-65 


7-88 


8-11 


8-46 


8-80 


9*13 


9-44 


9-72 


9-96 


10*44 


10-92 


11*34 


32 


7-80 


8-05 


8-30 


8-66 


9-00 


9-34 


9-64 


9-94 


10-28 


10-67 


11-16 


11*58 


33 


7-95 


8-22 


8-49 


8-85 


9-20 


9-54 


984 


10-15 


10-50 


10 89 


11*89 


11-82 


34 


8-10 


8-39 


8-68 


9-04 


9-40 


9-74 


10-04 


10-35 


10*71 


11*12 


11*62 


12*06 


35 


8-25 


8-56 


8-87 


9-23 


9-59 


9-94 


10-24 


10-55 


10*92 


11*34 


11*85 


12*29 


36 


8-40 


8-73 


9-06 


9-42 


9-79 


10-14 


10-44 


10-75 


11*13 


11-56 


12-OS 


12*53 


3T 


8 - 55 


8-90 


9-25 


9-61 


9-99 


10-34 


10-64 


10-95 


11*34 


11*78 


12-31 


12*76 


33 


8-70 


9-07 


9-44 


9-80 


10-18 


10-54 


10-84 


11-15 


11*55 


12-00 


12*54 


12*99 


I 39 


8-35 


9-24 


9-62 


9-89 


10-37 


10*73 


11-04 


11*35 


11*76 


12-23 


12*77 


13-23 


40 


9-00 


9-40 


9-80 


10-18 


10-56 


10*90 


11-24 


11*56 


11*88 


12*46 


12-98 


13*43 


41 


9-19 


9-59 


9-99 


10-37 


10-75 


11*08 


11-41 


11*73 


12-05 


12*67 


13*22 


13*71 


42 


9-38 


9-78 


1017 


10-56 


10-94 


11-25 


11-57 


11*89 


12*22 


12-87 


1346 


13-94 


43 


9-57 


9-97 


10-36 


10-75 


11-13 


11-42 


11-72 


12*05 


12*39 


13-08 


13-70 


14*16 


44 


9-76 


10-15 


10-55 


10-94 


11-32 


11-59 


11-S7 


12-21' 


12-56 


13-28 


13*93 


14*38 


45 


9-94 


10-33 


10-74 


11-13 


11-51 


11-76 


12-02 


12*37 


12-73 


13*49 


14*17 


14-60 


46 


10-12 


10-51 


10-93 


11-32 


11*70 


11-93 


12-17 


12-53 


12-90 


13*69 


14*40 


14*82 


48 


10-48 


10-S7 


11-30 


11-70 


12*08 


12-27 


12-47 


12*85 


13-24 


1410 


14-87 


15*26 


50 


10-86 


11-27 


11-68 


12-07 


12*46 


12-61 


12-76 


13*17 


13-58 


1454 


15-34 


15*70 


52 


11-20 


11-63 


12-06 


12-44 


12*84 


13-03 


13-22 


13*63 


14-04 


14-94 


15-71 


16*14 


54 


11-54 


11-98 


12-42 


12-81 


13-22 


13-45 


13-68 


14-09 


14*49 


15-34 


16-08 


16-5-5 


56 


11-88 


12-32 


12*78 


13-18 


13-60 


13-87 


14-15 


14-55 


14*95 


15-74 


1045 


16*98 


58 


12-20 


12-66 


13-15 


13-56 


13-98 


14-29 


14-62 


15-01 


15*41 


16*14 


16-82 


17*41 


60 


12-52 


13-01 


13-51 


13-94 


14-37 


14-73 


15-10 


15-47 


15*84 


16-54 


17*19 


17-82 


C2 


12-96 


13-41 


13-89 


14-32 


14-73 


15-10 


15-48 


15-85 


16-22 


16-92 


17*59 


18-23 


64 


13-38 


13-81 


14-26 


14-69 


15-09 


15*57 


15-S6 


16-23 


16-60 


17-31 


17*99 


18-64 


66 


13-80 


14-21 


14-63 


15-05 


15-45 


15-94 


1623 


16-61 


16-98 


17*70 


18*38 


19*05 


68 


14-22 


14-61 


15-00 


15*41 


15-81 


16-31 


16-60 


16-98 


17-36 


18*09 


18*78 


19-45 


70 


14-64 


15-01 


15-38 


15-77 


1617 


16-57 


16-97 


17*35 


17*74 


18-48 


19*18 


19*85 


72 


15-02 


15-39 


15-75 


16-08 


16-56 


16-93 


17-30 


17-71 


18-12 


18-86 


19*53 


20*26 


74 


15-40 


15-77 


16-12 


16-46 


16-95 


17*29 


17-63 


18*06 


18-49 


19*24 


19-97 


2067 


76 


15-78 


16-15 


16-49 


16-84 


17-34 


17-65 


17*97 


18-41 


18-86 


19-62 


20-85 


21-07 


78 


1616 


16-53 


16-S7 


17-22 


17-73 


18-01 


18-30 


18*76 


19-23 


20-00 


20*73 


21-48 


80 


16-54 


16-89 


17*24 


17-63 


18-12 


18-37 


18-61 


19*11 


19*61 


20*38 


21*11 21-88 


82 


16-94 


17-20 


17-60 


18-04 


18-47 


18-75 


19*03 


19-51 


19-99 


20*76 


21-51 


22*27 


& 


17-32 


17-65 


17-98 


18-40 


1S-82 


19-13 


19*45 


19*90 


20-36 


21-14 


21*89 


22*65 


| 80 


17-71 


18-03 


18-36 


18-76 


19-17 


19-51 


19*85 


20-29 


20.73 


21*52 


22-28 


23-03 


i 88 


18-09 


18-41 


18-73 


19-12 


19-52 


19-89 


20-26 


2063 


21-10 


21*89 


22*67 


23-41 


90 


18-47 


18-79 


19-11 


19-49 


19-87 


20-27 


20-67 


21*07 


2147 


22*26 


23*03 


23*79 


92 


18-87 


19-18 


19-49 


19*87 


20-25 


20-64 


21*04 


21-44 I21-S4 


22-64 


23*41 


24*17 


94 


19-26 


19-56 


19-87 


20*25 


•20-62 


21-01 


21*41 


21*81 22*21 


23-01 


23-79 


24-55 


96 


19-64 


19-94 


20-25 


20*63 


20-99 


21-38 


21-73 


22-18 22*48 


23 38 


24*16 


24*93 


98 


20-02 


20-32 


20-62 


21*00 


21*36 


21-75 


22-15 


22-55 22-S5 


23-75 


24-53 


25-31 


100 


20*40 


20-70 


21-00 


21*36 j 21*73 


22-12 


22-52 


22-92 123*32 


24-12 


24.90 


25-68 



DIMENSIONS OF THE CRANK. 



297 



THICKNESS OF TFEB OF CRANK:, SUPPOSING IT TO BE CONTINUED 
TO PADDLE-SHAFT CENTRE. 



c.E 

a" 
20 


LENGTH OP STROKE IN FEET. 


2 
2-G4 


2^ 


3 


3^ 


4 
3"23 


4, L 
8-36 


5 
3-49 


5* L 

3-59 


6 
8-70 


n 

3-39 


8 

4-06 


9 

4-22 


2-72 


2-31 


3-02 


21 


2-75 


2-34 


2-93 


3T4 


3-34 


3-47 


3-61 


3-71 


3-82 


4-02 


419 


4-36 


22 


2-S6 


2-96 


8-05 


3-25 


8'45 


3-58 


3-73 


3-83 


8-94 


4-14 


4-32 


4-50 


23 


2-97 


8-03 


3-17 


3-36 


3-56 


8-69 


3'S5 


3-95 


4-06 


4-26 


445 


4-64 


24 


3-03 


3-19 


3-29 


3-47 


8-67 


8-SO 


3-97 


4-07 


4-18 


4-38 


4-5S 


4-77 


25 


3-19 


S-30 


3-41 


3-58 


3-73 


3-91 


4-09 


4-19 


4-30 


4-50 


4-71 


4-90 


26 


330 


3-41 


3-52 


3-69 


8-39 


4-02 


4-20 


4-30 


4-42 


4-62 


4-84 


5-03 


27 


3 41 


3-52 


3-03 


3-80 


4-00 


4-13 


4-31 


4-41 


4-53 


4-74 


4-97 


5-16 


28 


352 


3-63 


3-74 


8-91 


4T0 


4-24 


4-42 


4-52 


4-64 


4-S6 


5-10 


5-29 


29 


3-63 


3-74 


3-85 


4-02 


4'20 


4-85 


453 


4-63 


4-75 


4-98 


5-22 


5-42 


30 


3-75 


8-85 


3-96 


4-13 


4-30 


4-46 


4-62 


4-74 


4-87 


5-11 


5-34 


5-55 


31 


3-83 


3-94 


4-06 


4"23 


4-39 


4-56 


4-72 


4-85 


4-98 


5-23 


5-46 


5-67 


32 


3-91 


4-03 


4T6 


4-33 


4-49 


4-66 


4-82 


4-96 


5-09 


5-34 


5-57 


5-79 


33 


3-99 


4-12 


4-26 


4-43 


4-59 


4-76 


5-92 


5-07 


5-20 


5-45 


5*69 


5-91 


£ 


4-07 


4-21 


4-36 


4-53 


4-69 


4-86 


5-02 


518 


5-31 


5 57 


5-80 


6-03 


35 


4-15 


4-80 


4-45 


4-62 


4-79 


4-96 


512 


5-23 


5-42 


5-68 


5-92 


6-15 


36 


4-22 


4-38 


4-54 


4-71 


4'S9 


5-06 


5*22 


5-38 


5-53 


5-79 


6-03 


6-27 


37 


4-29 


4-46 


4-63 


4-80 


4*93 


5-16 


5-32 


5-48 


5-64 


5-90 


615 


6-39 


£ 8 


4-36 


4-54 


4-72 


4-S9 


5-03 


5-26 


5-42 


5-58 


5-74 


6-01 


6-26 


6-51 


39 


4-43 


4-62 


4-81 


4-98 


5*13 


5-36 


5-52 


5'68 


5-84 


6-12 


6-33 


6-62 / 


40 


4-50 


4-70 


4-90 


5-09 


5*2S 


5-45 


5-62 


5-78 


5-94 


6-23 


6-49 


6-74 ) 


41 


4-60 


4-80 


5-C0 


5T9 


5-33 


5-54 


5*70 


5-86 


6-03 


6-34 


6-61 


6-86 | 


42 


4-70 


4-90 


5T0 


5-29 


5*48 


5-03 


5-78 


5-94 


611 


645 


6-73 


6-97 


43 


4-SO 


5-00 


5-20 


5-39 


5-53 


5-72 


5-86 


6-02 


6-20 


6-56 


6-S5 


7-08 


44 


4-89 


5-09 


5-30 


5-49 


5-68 


5-81 


5-94 


610 


6-28 


6-67 


6-97 


719 


45 


4-98 


5.18 


5-39 


5-58 


5'7S 


5-90 


6-02 


618 


6-37 


6-77 


7-09 


7-30 


46 


5-07 


5-27 


5-48 


5-67 


5-87 


5*98 


610 


626 


645 


6-87 


7-21 


7-41 


48 


5-25 


5-45 


5-66 


5-85 


6'05 


614 


6'26 


6-42 


6-62 


7-07 


7-45 


7-63 


50 


5-43 


5-63 


5-84 


6-03 


6-23 


6-30 


6-38 


6-58 


6-79 


7-27 


7-67 


7-85 


52 


5-61 


5-S1 


6-03 


6-23 


6*43 


6-52 


6-62 


6-81 


7-03 


7-47 


7-87 


8-07 


64 


5-73 


5-98 


6-21 


6-42 


6-63 


6-74 


6-S6 


7-04 


7-27 


7-67 


8-05 


8-29 


56 


5-94 


6-14 


6-39 


6-60 


6-82 


6-96 


7T0 


7-27 


7-50 


7-S7 


8-23 


8-51 


5S 


6T0 


6-30 


6-57 


6-7S 


7-00 


7T6 


7-33 


7*50 


7-72 


8-07 


811 


S-71 


60 


6-26 


6-50 


6 - 75 


6-96 


71S 


7-36 


7-55 


773 


7*92 


S-27 


8-59 


S-91 


62 


6-4S 


6-71 


6-95 


7-16 


7-36 


7-56 


7'75 


7-93 


811 


8-47 


8-T9 


9-11 


64 


6-70 


6-91 


7-15 


7-34 


7-54 


7-74 


7-94 


813 


8-30 


8-65 


8-97 


9-31 


66 


6-92 


7T0 


7-83 


7-52 


7-72 


7-92 


8-12 


8-31 


8-49 


8-84 


9-17 


9-52 


68 


742 


7-30 


7 51 


7-70 


7-90 


8-10 


8-30 


8-49 


8-68 


9-04 


9-37 


9-72 


70 


7-32 


7-50 


7-69 


7-88 


8-03 


8-28 


8-4S 


8-67 


8-87 


9-24 


9-59 


9-92 


72 


7-51 


7-69 


7-S7 


8-07 


8-28 


8-46 


8-66 


8-85 


9-07 


9-48 


9-83 


1012 


74 


7-70 


7-S8 


S-06 


8-26 


847 


864 


8-82 


9-03 


9-26 


9-71 


10-06 


10 32 


76 


7-89 


S-07 


8-25 


8-46 


8-67 


8-82 


8-98 


9-21 


914 


9-93 


10-28 


10-53 


78 


8-08 


8-26 


8-44 


8-65 


8-S7 


9-00 


9T4 


9-39 


9-62 


1016 


10-51 


10 73 


80 


8-27 


8-44 


8-62 


8-84 


9-06 


913 


9-30 


9-55 


9-80 


10-37 


10-75 


10 94 


82 


8-46 


8-63 


8-81 


9-02 


9*24 


9-37 


9-41 


9-75 


9-99 


10-53 


10-91 


1113 


84 


8-65 


8-82 


9-00 


9-20 


9-42 


9-56 


9-72 


9-94 


1013 


10-69 


11-07 


11-31 


86 


8-S5 


9-01 


913 


9-38 


9-60 


9-74 


9-92 


10-13 


10.36 


10-84 


11.23 


11-50 


S3 


9-04 


9-20 


9-36 


9 56 


9-75 


9-93 


10-13 


10-33 


10-54 


10-98 


11-37 11-69 


90 


923 


9-39 


9-55 


9-74 


9-93 


10T3 


10-33 


10-53 


10-73 


1113 


11-51 


11 -S9 


92 


9-43 


9-53 


9-74 


9-93 


10-11 


10-32 


10-51 


10-71 


10-92 


11-32 


11-69 


12-08 


94 


9-63 


9-78 


9-93 


10-12 


10-30 


10-50 


10-70 


10-90 


11-11 


11-50 


11-S9 


12-27 


96 


9-82 


9-97 


10-12 


10-30 


10-49 


10-69 


10-89 


11-09 


11-29 


11-68 


12-03 


12-46 


98 


10-01 


10-16 


10-31 


10-49 


10-63 


10-SS 


11 -OS 


11-27 


11-48 


11-37 


12-27 


12-64 


100 


10-20 


10-35 


10-50 


10-68 


10-36 


11-06 


11-26 


11-46 


11-66 


12-06 


12-45 


12-84 



13* 



298 



PROPORTIONS OF STEAM-ENGINES. 



THICKNESS OP LAKGE EYE OF CKANK. 



"5.3 
** s °° 

<U Co 








LENGTH OF STROKE 


IN FEET. 


• 






5° 


2 


1-80 


3 

1-S9 


8* 

1-97 


4 

2-05 


*i 


5 

2-19 


w 


6 


7 

2-44 


8 
2-55 


9 

2-64 


20 


1-71 


212 


2-25 


232 


21 


1-77 


1-87 


1-95 


2-04 


2-13 


2-20 


2-27 


2-32 


240 


2-52 


264 


275 


22 


1-83 


1-93 


2-01 


2-11 


2-20 


2-28 


2-35 


2-39 


2-48 


2-60 


272 


2-86 


23 


1-89 


1-99 


2-07 


2-18 


2-28 


2-36 


243 


246 


2-56 


2-68 


2-80 


2-97 


24 


1-95 


2-06 


2-14 


2-25 


2-35 


2-44 


2-51 


2-54 


2-64 


276 


2-88 


3-08 


25 


2-01 


2-12 


2-21 


2-32 


2-43 


252 


2-59 


2-62 


2 72 


2-84 


2-96 


3-18 


26 


2-07 


249 


2-28 


2-39 


2-50 


2-59 


2 66 


270 


2-80 


2-92 


3-04 


3-28 


21 


2-13 


2-25 


2-35 


2 46 


2-58 


2-66 


273 


278 


2-87 


2-99 


312 


3-33 


23 


2-19 


2-31 


2-42 


2-53 


2-65 


2-73 


2-SO 


2-86 


2-94 


3-06 


3-20 


3-43 


29 


2-25 


2-37 


2-49 


2-60 


2-73 


2-80 


287 


2-94 


3-01 


3-13 


3-28 


3-58 


30 


2-30 


2-43 


2-56 


2-68 


2-80 


2-87 


294 


3-01 


308 


318 


3-36 


3-6S 


31 


2-36 


2-50 


2-63 


2-74 


2-S7 


2-94 


3-00 


3-07 


845 


326 


3-43 


373 


32 


2-42 


2-56 


2-69 


2-80 


2-94 


3-01 


3-06 


3-13 


322 


3-34 


3-50 


878 


33 


2-49 


2-62 


2-75 


2-S6 


300 


3-08 


3 12 


3"20 


3-29 


3-41 


3 57 


3-83 


34 


2-55 


2-69 


2-81 


2-92 


3-06 


315 


3-1S 


327 


336 


3-48 


3 64 


3-88 


35 


2-61 


2-75 


2-S7 


2-98 


3-12 


3-22 


3-25 


3 34 


343 


3-55 


371 


4-93 


36 


2-67 


2-81 


2-93 


3-04 


3-18 


3-29 


3 32 


341 


8-50 


3-62 


878 


4-98 


37 


2-74 


2-87 


299 


3-10 


3-24 


3-35 


339 


348 


3-57 


3-69 


3-85 


4-03 


38 


2-81 


2-94 


3-05 


3-16 


3*30 


341 


3 46 


355 


3-64 


377 


3-91 


4-07 


39 


2-87 


3-00 


3-11 


3-22 


3-36 


347 


3-53 


3-62 


371 


3-84 


3-97 


441 


40 


2-93 


3-05 


347 


3-29 


3-42 


351 


3-60 


3-69 


378 


3-90 


4-03 


445 


41 


3-03 


3-13 


3-24 


3-37 


3-49 


3-58 


3-67 


375 


3-85 


3-98 


4-11 


4-24 


42 


3-13 


3-22 


3-31 


3-45 


3-56 


3-65 


374 


3-81 


392 


4-06 


4-19 


4-32 


43 


3-23 


3-30 


3-39 


3-53 


3-63 


3-72 


381 


3-37 


399 


4-14 


4-27 


440 


44 


3-33 


3-39 


3-47 


3-60 


3-69 


3-79 


3-88 


3'94 


4-06 


4-22 


4-35 


448 


45 


3-43 


3-47 


3-55 


367 


3-75 


3-86 


3-95 


4-01 


4-13 


4-30 


4-43 


4-56 


46 


3-53 


3-56 


3-63 


3-74 


3-81 


3-93 


4-02 


4-08 


419 


4-38 


4-51 


4-64 


48 


3-73 


3-73 


3-79 


3-88 


3-93 


4-06 


415 


4-22 


4-32 


4-52 


4-66 


4-80 


50 


3-93 


3-92 


3-95 


4-02 


4-09 


4-18 


4-27 


4-36 


446 


4-64 


4-81 


4-96 


52 


4-10 


4-05 


4 09 


4-18 


4-24 


4-31 


4-41 


448 


4-60 


478 


4-95 


540 


54 


4-28 


4-19 


4-24 


4-33 


4-38 


4-45 


4-55 


4-61 


474 


4-92 


5-09 


5-24 


56 


4-45 


4-33 


4-40 


447 


4-52 


4-59 


4-69 


475 


4-88 


5-06 


5-23 


5-38 


58 


4-62 


4-46 


4-56 


4-61 


4-66 


4-73 


4-83 


4-89 


5-00 


5-19 


5-37 


5 52 


60 


4-79 


4-60 


4-72 


4-76 


4-80 


4-87 


4-95 


5-03 


512 


5-21 


549 


5-66 


62 


4-98 


4-80 


4-88 


4-90 


4-96 


5-02 


5-09 


517 


5-26 


545 


5'63 


5-80 


64 


5-16 


5-00 


5-04 


5-03 


510 


5-16 


5-23 


5-31 


5-40 


5-59 


577 


5'94 


66 


5-34 


5-20 


5-20 


5-15 


524 


5-30 


5-37 


5*45 


5-54 


573 


5-90 


6-08 


6S 


5-52 


5-40 


5-35 


5-27 


538 


544 


5 51 


5-59 


5-68 


5-87 


6-04 


6-22 


70 


5-70 


5-00 


5-49 


5-40 


5-52 


5-58 


5-64 


572 


5-80 


5-98 


616 


6-35 


72 


5-90 


5-78 


5-67 


5-56 


5-68 


574 


580 


5-S6 


5-94 


612 


6-30 


6-49 


74 


6-09 


5-96 


5-84 


5-72 


584 


5 89 


595 


6-00 


6-08 


6-26 


644 


6-63 


76 


6-27 


6-14 


6-00 


5-88 


5-98 


603 


6-09 


6-14 


6-22 


6-40 


6-58 


675 


78 


6-46 


6-32 


6-16 


6-02 


6-12 


617 


623 


6-28 


636 


6-53 


671 


6-87 


80 


6-66 


6-49 


6-32 


6-16 


6-26 


6-31 


637 


6-43 


6-50 


6-66 


6-84 


7-02 


82 


6-S6 


6-69 


6-50 


6-35 


6-42 


6 46 


6-51 


657 


6-64 


6-80 


6-98 


7-16 


84 


7-06 


6-88 


6-6S 


6-54 


6-58 


6-61 


6-65 


671 


678 


6-94 


712 


7-30 


86 


7-27 


7-06 


6-84 


6-73 


6-74 


676 


679 


6-35 


6-90 


7-08 


72 6 


7-44 


88 


7-47 


7-24 


7-00 


6-92 


6-90 


6-91 


6-93 


6.99 


7-02 


7*21 


7-39 


7-57 


90 


7-67 


7-42 


713 


711 


7-05 


7-06 


7-08 


711 


714 


7-34 


7-51 


7-69 


92 


7-8S 


7-62 


7-37 


7-29 


7-21 


7-22 


7-23 


7-27 


7-30 


7-48 


7*65 


7*83 


94 


8-09 


7-31 


7-55 


7-47 


7-37 


7'38 


7-38 


7-43 


745 


7-62 


779 


7-97 


96 


8-31 


7-99 


7-73 


7-65 


7-53 


7-54 


753 


7'59 


7-60 


776 


7 93 


841 


98 


8-52 


8-17 


7-91 


7-81 


7-69 


770 


7-68 


773 


776 


7 90 


8-06 


8-24 


100 


8-72 


840 


8-09 


7-97 


7-86 


7-84 


7-83 


7 87 


7-91 


8-04 


8-19 


8-30 







DIMENSIONS OF AIR-PUMP AND PISTON ROD. \ 


>U9 


•a 

s 

O 

O aa 


p. 

a 

3 
P< 

-** « 


DIMENSIONS OF THE SEVERAL PAKTS OF PISTON EOD IN INCHES. 




s 

£ g 




i. a 

C 1- £ 


i a 

lii 

QA< <B 

o u £ 


U I 

6 a 
ts— ■ 

si 


fr4 1 

"Ssn 
1-3 

.^P-l S3 


-8o 

o g 
•S6-S 
*2"S ° 


o> g 

G = $ 

•8" 8 


5 o 

O; 


o o 
S Ss h 


».S a. 

CMC 


* = 


= .2 


"S 


°.S 


'S*.2o 


SSo 


S*S-S 


.5^ 


•POO 


3«BO 


C^d 


36S 


|S 8 


3 


3 


5 


>2 


S 


s 


s 


a 


o 


H 


ft*" 


H 


O 


20 


12-0 


2-0 


4-0 


1-90 


1-80 


2-80 


2-80 


211* 


•42 


1-70 


•70 


1-34 


21 


12-e 


21 


4-2 


1-99 


1-89 


2-94 


2-41 


2-21 


•44 


1-78 


•73 


1-40 


22 


18-2 


2-2 


4-4 


2-09 


1-98 


3-08 


2-53 


2-32 


•46 


1-87 


•77 


3 47 


23 


13-8 


2-3 


46 


2i8 


2-07 


3-22 


2-64 


2-42 


•48 


1-95 


•so 


1-53 


24 


14-4 


2-4 


4-8 


2-23 


2-16 


3-36 


2-76 


2 53 


•50 


2-04 


•84 


1-60 


25 


15-0 


2-5 


5-0 


2-37 


2-25 


3-50 


2-S7 


2-63 


•52 


212 


•87 


1-67 


26 


15-6 


2-6 


5-2 


2-47 


2-34 


3*64 


2 99 


2-74 


•54 


2-21 


•90 


1-73 


27 


16-2 


2-7 


5-4 


256 


243 


3-78 


311 


2-84 


•57 


2-29 


■94 


1-80 


28 


36-8 


2-8 


5-6 


2-66 


2-52 


392 


3-22 


2-95 


•59 


2-38 


•97 


1-87 


29 


17-4 


2-9 


58 


2-75 


2-61 


4-06 


3 34 


3-05 


•61 


246 


1-00 


1-94 


30 


1S-0 


3-0 


6-0 


2-S5 


2-70 


4-20 


345 


316 


•63 


2-55 


1-04 


2-00 


31 


1S-6 


31 


6-2 


2-94 


2-79 


4-34 


3-57 


3-26 


•65 


2-63 


1-07 


2-07 


32 


19-2 


3-2 


6-4 


3-04 


2-83 


448 


3-68 


3-37 


•67 


2-72 1 


110 


214 


33 


19-8 


3-3 


6-6 


3-13 


297 


4-62 


3 -SO 


8*47 


•69 


2-SO 


114 


2-21 


34 


20-4 


3-4 


6-8 


3-23 


3-06 


4-76 


3-91 


8-57 


•71 


2-89 1 


119 


2-27 


35 


21-0 


3-5 


7-0 


3-32 


3-15 


4-90 


4-02 


3-67 


•73 


2-97 


1-22 


2-33 


36 


21-6 


3-6 


7-2 


3-42 


3*24 


5-04 


414 


3-78 


•75 


3-06 


1-26 


2-40 


37 


22-2 


3-7 


7-4 


3-51 


3 33 


5-13 


425 


3-88 


•78 


314 


1-29 


2-47 


38 


22*8 


3-8 


7-6 


3-61 


342 


5-32 


436 


3-99 


•80 


3-23 


133 


2-54 


39 


23-4 


3-9 


7-8 


3-70 


3-51 


5-46 


4-48 


4-09 


•82 


3-31 


1-36 


2-60 


40 


24-0 


4-0 


8-0 


3-80 


3-60 


5-60 


4-59 


4-20 


•84 


3-40 


140 


2-67 


41 


24-6 


4-1 


8-2 


3-89 


3-69 


5-74 


4-70 


4-30 


•86 


3-48 


1-43 


2-74 


42 


25-2 


4-2 


8-4 


3-99 


3-78 


5-88 


4-82 


4-41 


•89 


8-57 


1-47 


2 -SI 


43 


25-8 


4-3 


8-6 


4-08 


3-87 


6-02 


4-93 


4-51 


•91 


8-65 


1-50 


2-87 


44 


26-4 


4-4 


8-8 


4-13 


3-96 


6-16 


5-05 


4-62 


•93 


8-74 


1-54 


2-98 


45 


27-0 


4-5 


9-0 


4-27 


4-05 


6-80 


5-17 


4-72 


•95 


3-82 


1-57 


3-00 


46 


27-6 


4-6 


9-2 


4-37 


4-14 


6-44 


5-28 


4-S3 


•97 


3-91 


1-61 


3-07 


48 


28-8 


4-3 


9-6 


4-56 


4-32 


6-72 


5-51 


5-04 


1-02 


4-08 


1-68 


3-20 


50 


30-0 


5-0 


10-0 


4-75 


4-50 


7-CO 


5-74 


5-25 


1-07 


4-25 


1-75 


3-83 


52 


31-2 


5-2 


104 


4-94 


4-68 


7-28 


5-97 


5-46 


111 


4-42 


182 


3-47 


54 


82-4 


5-4 


10-8 


5-13 


4-86 


7-56 


6-21 


567 


115 


4-59 


1-89 


3-60 


56 


33-6 


5-6 


11-2 


5-32 


5-04 


7-84 


6-44 


5-88 


119 


4-77 


1-96 


3-74 


53 


34*8 


5-8 


11-6 


5-51 


5-22 


8-12 


667 


6-09 


123 


4-94 


2-03 


3-88 


60 


36-0 


6-0 


12-0 


5-70 


5-40 


840 


6-90 


6-30 


1-27 


511 


210 


4 01 


62 


37-2 


6-2 


12-4 


5-S9 


5-58 


8-68 


7-13 


6-51 


1-31 


5-2S 


217 


4-14 


64 


3S4 


6-4 


12-8 


6-08 


576 


8-96 


7-36 


6-72 


1-35 


5-45 


2-24 


4-27 


66 


39-6 


6-6 


13-2 


6-27 


594 


9-24 


7-59 


6-93 


1-39 


5-62 


2-31 


4-40 


6S 


40-3 


6-8 


13-0 


646 


612 


9-52 


7-82 


7-14 


1-40 


5-79 


2 38 


4-53 


70 


420 


7-0 


14-0 


6-65 


6-30 


9-80 


8-05 


7-35 


1-47 


5-96 


244 


4-67 


72 


432 


7-2 


144 


6-S4 


648 


10-08 


8-28 


7-56 


1-51 


613 


2-51 


4-80 


74 


44-4 


7-4 


14-8 


7-03 


6 66 


10-36 


8-51 


7-77 


1-55 


6-30 


2-53 


4-93 


76 


45-6 


7-6 


15-2 


7-22 


6-84 


10 64 


8-74 


7 9S 


1-67 


646 


2-66 


5-07 


73 


46-8 


7-3 


15 • 


7-41 


7*02 


10 92 


8-97 


8-19 


1-70 


6-63 


2-73 


5-20 


80 


48-0 


8-0 


160 


7-CO 


7-20 


11-20 


9-20 


8-40 


1-73 


6-80 


2-80 


5-33 


82 


49-2 


8-2 


16-4 


7-79 


7-38 


11-43 


9-43 


3-61 


1-76 


6-97 


2-87 


5-47 


84 


50-4 


S-4 


16-8 


7-98 


7-56 


11-76 


9-66 


8-S2 


1-79 


714 


2-94 


5-60 


86 


51*6 


8-6 


17-2 


8-18 


7-74 


12-04 


9 89 


9-03 


1-S2 


7-31 


3-01 


5-73 


88 


52-8 


8-8 


17-6 


8-37 


7-92 ,12-32 


10-12 


9-24 


1-85 


748 


3 08 


5-^7 


9C 


54-0 


9 


ISO 


S-56 


8-10 12-60 


10-34 


9-45 


1-89 


7-66 


315 


6-00 


92 


55-2 


9-2 


18-4 


8-75 


8-28 12-88 


110-57 


9-G6 


1-92 


7-83 


3-22 


644 


94 


56-4 


9-4 


18-8 


8-94 


846 13-16 


10-80 


9-87 


195 


8-00 


8-29 


6-27 


96 


57'6 


9-6 


19 "2 


9-12 


8-64 13-44 


1103 


10-08 


2-00 


817 


3-36 


6-41 


93 


5S-8 


9-8 


19-6 


9-32 


8-82 13-72 


11-26 


10-29 


2 05 


8-34 


3-43 


6-54 


100 


60-0 


10-0 


20-0 


9-50 


9-00 14-00 


11-49 


10-50 


2.11 


8-51 


3-50 


666 



300 



PROPORTIONS OF STEAM -UfGENES. 



5 
£ 


ceavk pix. 




CKAKK 






PIPE 


5 axd 


PA55A 


SES. 












%- ~ . jz 


•— 5— 


^ J ■ ; - £ — ^ 


-— . t- 5 


— z 


'- - X 


■Wjjj I =->> 


o 


Z £ 




= i 






- J: _= 


gj°e 








«_l 


r* 


"h = 


© 


-s "*• "S £ „ 2 


m 


=: - 




■"?•.-§ 


a 


-is . 


z x 
- = 


ft "3 

1 a i 


o s 

"5 z 

-5*"" 

6bS 


■5 = 5 
a 5 *■ 

gene 


£5 g2<3 

"5:_>> u 5 ~ 




- 

9 

E 

z 


s.5 
t» • -. 

~ a .- 

- i i 
= >=- 
.= ■- — 


z: a 


<~ — © 
o »- 


©— as 
z Z i 


> -i 

j'0| 


- 


Q 


J±l_ 




iJ - 


B 


5 


- 


< 


«-j 


- 


5-00 


20 


2"54 


3-20 


"l-26 


3-75 2-20 3-20 


2-69 


17 


315 


1-79 


21 


•_' . 


3-36 


l-?3 


3-93 ! 2-31 


3-36 


10 


3-32 


26 


3-49 


1-54 


5"23 


22 


3-12 


3-52 


1-40 


4-12 2-42 


3*52 


15 


3-94 


35 


354 


1-59 


5-45 


2" 


3-24 


3-63 


1-46 


4-31 


2-53 


8-65 


20 


4-56 


44 


4-15 


194 


5-66 


2-1 


3-35 


3-34 


1-52 


4-50 


2-64 


3*84 


25 


5-13 


53 


4-53 


1-99 


5-86 


;3 


3-52 


4-00 


1-53 


4-63 


2-75 


4-00 


30 


530 


62 


4*87 


2-04 


6*05 


- 


3-66 


4-16 


1-65 


457 


2-86 


4-16 


35 


6-42 


71 


5*22 


2-09 


6"24 


27 


3- SO 


4-32 


1-71 


5-06 


2-97 


4-32 


40 


7-04 


80 


5-58 


2-13 


6-42 


28 


3-94 


4-43 


1-77 


5-25 


■ 


4-45 


45 


7-66 


89 


5*91 


2-18 


6(50 


29 


4--S 


4-64 


153 


5-43 


3-19 


4-64 


50 


8*28 


98 


6-25 


2 23 


6-SO 


30 


426 


4-30 


1-90 


5-62 


3-30 


4-50 


55 


8-90 


107 


6-60 


2-29 


7-07 


31 


4-40 


496 


1-95 


5-51 


3 41 


4-96 


CO 


9-50 


116 


6-94 


2-32 


7-23 


32 


4-54 


5-12 


2-02 


5-99 


3-52 


512 


65 


10-06 


125 


7 29 


2-36 


7-39 


S3 


4-63 


5-23 


2-03 


6-13 


3-63 


5-23 


70 


10-56 


134 


7-63 


2-40 


7*55 


34 


4-53 


5-44 


214 


6-37 


3 '74 


5-44 


75 


10-96 


143 


7-93 


2-44 


7 '71 


85 


4-97 


5-60 


2-21 


6-56 


3-55 


5-60 


80 


11-31 


152 


8-32 


2 45 


7-57 


36 


511 


5-76 


2-27 


6-74 


3-96 


576 


S5 


11-61 


161 


8-67 


2-52 


8-03 1 


37 


5-36 


5-92 


2 -"'3 


6-93 


4-07 


5-92 


90 


11-89 


170 


9-01 


2-56 


8-19 


o? 


5-40 


6-03 


2-39 


7-12 4-13 


6*08 I 


95 


12-09 


179 


9-36 


2-60 


8-35 


39 


5'54 


6-24 


2-45 


7-31 


4-29 


6-24 


100 


12-19 


188 


9-70 


2-64 


8-51 


40 


5-69 


6-40 


2-52 


7-50 


4-40 


6-40 


105 


12-29 


197 


10-05 


2 68 


S66 


41 


5-33 


6-52 


2-58 


: 


4-51 


6-56 | 


110 


12 56 


2C6 


10-39 


2-72 


" 


42 


5-97 


6-63 


2-64 


r-87 


462 


6-72 ; 


115 


12*83 


215 


10-74 


2-76 


S-94 


43 


6-11 


6-34 


2-71 


8-05 


4-73 


6-53 


120 


13-10 


224 


1103 


2-50 


9-O8 


1 44 


6-25 


:• ) 


2-78 


8-24 


434 


7C4 , 


125 


13-37 


233 


11-43 


0i 4 


9 '22 


45 


6-39 


7-16 


2-34 


8*42 


4-95 


7-20 


130 


13-64 


242 


11-77 


- 58 


9*36 


46 


6-54 


7-32 


2 91 


8-61 


506 


7*36 


135 


13-91 


251 


12-12 


2-91 


950 


45 


•3-52 


7-63 


3-03 


- - 


5*28 


7-63 


140 


14-13 


260 


1246 


2-94 


9-63 


50 


7-11 


7-95 


3-16 


9*35 


5-50 


8-00 


150 


14-70 


273 


13-15 


3 00 


9-59 


52 


7-39 


3-27 


3 23 


9-72 


5-72 


8-32 


160 


15-20 


296 


1384 


3 07 


10-12 


54 


767 


B-39 


3-41 


10-09 


594 


8-64 


170 


1565 


314 


14-53 


3-14 


10-86 


56 


7 05 


S-9L 


3-53 


10-46 


616 


S-96 


150 


16-09 


332 


15-22 


3-20 


10-60 


5S 


8-24 


9-23 


8-66 


10-84 


6-33 


9-23 


190 


16-53 


850 


15-91 


3-26 


10 54 


60 


8-52 


9-54 


3-73 


11-20 


6-60 


9-60 


200 


16-97 


363 


16-60 


3-32 


11-06 


02 




9-86 


3-91 


11-57 


652 


9-92 


210 


17 39 


356 


17-29 


8-33 


11-29 


64 


9 09 


10-13 


4-04 


11-94 


7-04 


10-24 


220 


17-79 


404 


17-93 


3-44 


11-51 


66 


9-37 


10-50 


4-17 


12-31 


:•-•' 


10-56 


230 


1819 


422 


13-67 


3-50 


1173 


: 


9-65 


: 82 


4"29 


12-63 


7-49 


i :-:- 


240 


18-58 


440 


1936 


3-56 


11-95 


1 


994 


11-13 


4-42 


13-08 


"- 11*20 


250 


13-97 


453 


2005 


3-61 


1215 


72 


! -- 


11-45 


4-54 


13-44 


7-92 11 -52 


260 


19-34 


476 


2074 


3-66 


1285 


74 


10-50 


11-77 


4-67 


13-81 


5-14 11-84 


270 


19-70 


494 


21-43 


3-72 


12*55 


76 


10-79 


12*08 


4-79 


14-19 


S-36 1216 


2 : 2006 


512 


2212 


3-77 


12-75 


73 


11-07 


12-40 


4-91 


14-58 


5-53 12-45 


290 20-42 


530 


22-51 


3S2 


12-95 


80 


U*36 


12-72 


5-03 


14-94 


3*80 12-80 


300 


20-73 


543 


23-50 


3 83 


13-14 


-- 


11-64 '13-04 


5-16 


15-32 


. 13-12 


810 


2112 


568 


24-19 


3-93 


13-33 


S4 


11-93 13-85 


5-29 


15-69 


9 24 13-44 


320 


21-46 


584 


24-88 


3-98 


13-51 


86 


12-21 13-67 


5-41 


16-07 


9-46 13-70 


330 


21-80 


602 


25-57 


403 


13 69 


83 


i-2%5*) 13-99 


5-54 


16-44 


9-68 14-03 


§40 


2214 


620 


2626 


4-07 


13-57 


1 90 


12-79 14-30 


: ' 


16-82 


9 90 14*40 


' 22-46 : 


635 


26-95 


4-12 


14-05 


: 32 


13 7 14-62 


5-80 


17-20 


10-12 14-72 


SCO 22-77 


656 , 


27-64 


416 


14-: 3 


! i'l 


13-35 14-94 


5-92 


17'53 


10-31 1 


370 23-09 ' 


674 


28-33 


4-21 


14-41 


! 93 


J3-5-1 15-26 


: 


17-95 


l0'5G 15-38 | 


380 23-40 


692 


29 02 


4-2-3 


14*f9 


! 93 


- ■ 2 15-53 


017 


13-37 10 7: :~^ : 


390 23-70 ' 


710 


29-72 


431 


1476 


1 100 
L 


■- 




6-30 


13-75 


1100 


1600 | 


400 


24 ] 


725 


30-41 


4-37 


14 92 



DIMENSIONS PROPER FOR LOCOMOTIVES. 30 1 

I may here repeat that the diameter of cylinder in inches is 
given in the first vertical column, beginning at 20 inches and 
ending at 100 inches, while the length of the stroke in feet is 
given in the first horizontal column, beginning with 2 feet and 
ending with 9. If, therefore, we wish to find the dimension 
proper for any given engine, of which we must know the diam- 
eter of cylinder and length of stroke, we find in the first vertical 
column the given diameter in inches, and in the first horizontal 
column the given length of stroke in feet ; and where the vertical 
column under the given stroke intersects the horizontal column 
opposite the given diameter, there we shall find the required 
dimension.* 

LOCOMOTIVE ENGINES. 

It would be a mere waste of time and space to recapitulate 
rules similar to the foregoing as applicable to locomotive en- 
gines, since the strengths and other proportions proper for loco- 
motives can easily be deduced by taking an imaginary low pres- 
sure cylinder of twice the diameter of the intended locomotive 
cylinder, and therefore of four times the area, when the propor- 
tions will become at once applicable to the locomotive cylinder 
with a quadrupled pressure, or 100 lbs. on the square inch. In 
locomotive engines the piston rod is generally made ^th of the 
diameter of the cylinder, whereas by the mode of determining 
the proportions that is here suggested it would be -|th. But 
piston rods are made of their present dimensions, not so much 
to bear the tension produced by the piston, as to bear the com- 
pression when they act as a pillar ; and properly speaking the 
proportionate diameter should diminish with every diminution 

* For screw or other short-stroke engines working at a high speed, the strengths 
of shafts given in the foregoing tables should be somewhat increased, and the 
length of bearing at least doubled. In some recent screAV engines an irregular 
motion of the engine has been perceived, owing to the elasticity of the shaft. For 

3 //i>\ 3 i>2 x IT 
such engines a correspondent suggests the formula i/ ! — \ + -- _ ( ij ain . 

eter of journal in inches; where d = diameter of the cylinder In inches and e = 
radius of crank in inches. 



302 PROPORTIONS OF STEAM-ENGINES. 

in the length of the stroke. In very short cylinders a proportion 
of -^ of the diameter of the cylinder would suffice in the case 
of low pressure engines, which answers to a th of the diameter 
in locomotives where the stroke is always very short. But in 
high pressure engines of any considerable dimensions, carrying 
IOC lbs. on the inch, the diameter of the piston rod should he 
•5-th of the diameter, answering to -^th of the diameter in low 
pressure engines of the common total pressure of 25 lbs. pos 
Bquare inch. 



CHAPTER V. 

PROPORTIONS OF STEAM-BOILERS. 

In proportioning boilers two main requirements have to be 
kept in view : 1st. The provision of a sufficient quantity of grate- 
bar area to burn — with the intended velocity of the draught — 
the quantity of coals required to generate the necessary quantity 
of steam ; and 2d. The provision of a sufficient quantity of heat- 
ing surface in the boiler, to make sure that the heat will be prop- 
erly absorbed by the water, and that no wasteful amount of 
heat shall pass up the chimney. Even the quantity of heating 
surface, however, proper to be supplied for the evaporation of a 
given quantity of water in the hour will depend to some extent 
upon the velocity of the draught through the furnace: for upon 
that velocity will depend the intensity of- the heat within the 
furnace, and upon the intensity of the heat will depend the 
quantity of water which a given area of surface can evaporate. 
The first point therefore to be investigated is the best velocity 
of the draught, and the circumstances which determine that 
velocity. Here, too, there are two guiding considerations. The 
first is, that if the velocity of the draught be made too great, the 
small coals or cinders will be drawn up into the chimney and 
be precipitated as sparks, causing in many cases serious annoy- 
ance. The second consideration is, that the temperature of the 
escaping smoke should be as low as possible, and should in no 
case exceed 600°. While, therefore, it is desirable in land and 
marine boilers to have a rapid draught through the furnace — 
such as is produced in locomotives by the blast-pipe — in order 
that the heat maybe sufficiently intense to enable a small amount 
of surface to accomplish the required evaporation, it is at the 



804 PROPORTIONS OP STEAM-BOILERS. 

same time inadmissible to have sncli a rapid draught in the 
chimney as will suck up and scatter the small particles of the 
coal; nor is it desirable that the velocity of the air passing 
through the grate-bars should be so great as to lift small pieces 
of coal or cinder and carry them into the flues. No furnace has 
yet been constructed which reconciles the conditions of a high 
temperature with a moderate velocity of the entering air : but 
such a furnace may be approximated to by making the opening 
through the fire-bridge very small, and by insuring the necessary 
flow of air through these small openings by the application of a 
horizontal steam -jet at each opening; as by this arrangement a 
high temperature may be kept up in the furnace, at the same 
time that the contraction of the area through or over the bridge 
will not so much impair the draught as to prevent the requisite 
quantity of coal from being burnt. 

The exhaustion which a chimney produces is the effect of the 
greater rarity of the column of air within the chimney than that 
of the air outside. If the air be heated until it is expanded to 
twice its volume, then, its density being half of what it was 
before, each cubic inch of the hot air will weigh only half as 
much as a cubic inch of cold air ; and if the hot air be enclosed 
in a balloon, it will ascend in the cold air with a force of ascent- 
equal to half the w eight of the balloon full of cold air. As water 
is about 773 times heavier than air at the freezing-point, it will 
require 773 cubic inches of air, heated until they expand to twice 
their volume, to have ascensional force sufficient to balance a 
cubic inch of water : or if a syphon-tube be formed with a col- 
umn of water 1 inch high in one leg, it will require a column of 
the hot air 1546 inches (or nearly 129 feet) high, in the other 
leg, to balance the column of water 1 inch high. In other words, 
a chimney heated until the density of the smoke is only half that 
of the air entering the furnace, and which will be the case at a 
temperature under 600°, will, if 129 feet high, produce an ex- 
haustion of 1 inch of water. In land boilers the ordinary ex- 
haustion or suction of chimneys is such as would support a col- 
umn of from 1 to 2 inches of water. But in steam-vessels the 
height of the chimney is limited, and the deficient height has to 



PROPER HEIGHTS FOR CHIMNEYS. 305 

bo made up for by an increased area. In practice, the diameter 
of the chimney of a steam-vessel is usually made somewhat less 
than the diameter of the cylinder, there being supposed to be 
one chimney and two cylinders, with the piston travelling at the 
speed usual in paddle vessels. 

Boulton and "Watt's rule for proportioning the dimensions of 
(hs chimneys of their land engines is as follows : — 

BOULTON AND WATT'S RULE TOE FIXING- THE PEOPEE SECTIONAL 
AEEA OP A CHIMNEY OF A LAND BOILEE WHEN ITS HEIGHT 
IS DETEEMINED. 

Rule. — Multiply the number of pound? of coal consumed under 
the boiler per hour by 12, and divide the product by the square 
root of the height of the chimney in feet : the quotient is the 
proper area of the chimney in square inches at the smallest 
part. 

Example. — What is the proper sectional area of a factory 
chimney 80 feet high, and with a consumption of coal in the 
furnace of 300 lbs. per hour? 

Here 300 x 12 = 3,600 ; and divided by 9 (the square root 
of the height nearly) we get 400, which is the proper sectional 
area of the chimney in square inches. If therefore the chimney 
be square, it will measure 20 inches each way within. 

BOULTON AND WATT'S EULE FOE FIXING THE PEOPER HEIGHT OF 
THE CHIMNEY OF A LAND EOILEE WHEN ITS SECTIONAL AEEA 
IS DETEEMINED. 

Rule. — Multiply the number of pounds of coal consumed under 
the boiler per hour by 12, and divide the product by the sec- 
tional area of the chimney in square inches : square the quo- 
tient thus obtained, which will give the proper height of the 
chimney in feet. 

Example. — What is the proper height in feet of the chimney 
of a boiler which burns 300 lbs. of coal per hour, the sectional 
area of the chimney being 400 square inches? 

Here 300 x 12 = 3,600, which divided by 400 (the sectional 



006 PROPORTIONS OF STEAM-BOILERS. 

area) = 9, the square of "which is 81 ; and this is the proper 
height of the chimney in feet. 

These rules, though appropriate for land "boilers of moderate 
size, are not applicable to powerful boilers with internal flues, 
such as those used in steam-vessels, in which the sectional area 
of the chimney is usually adjusted in the proportion of 6 to 8 
square inches per norpinal horse-power. This will plainly appear 
from the following investigation: — - 

In a marine boiler suitable for a pair of engines of 110-horse- 
power, the area of the chimney, allowing 8 square inches per 
nominal horse-power, would be 880 square inches. Supposing 
the boiler to consume 10 lbs. of coal per nominal horse-power 
per hour, or say 10 cwt. (or 1120 lbs.) of coal per hour, and that 
the chimney was 46 feet high, then, by Boulton and "Watt's rule 
for land engines, the sectional area of the chimney should be 
1120 x 12 -5- V4G = 13,440 -5- say 7=1,920 square inches. This, 
it will be observed, is more than twice the area obtained by 
allowing a sectional area of 8 square inches per nominal horse- 
power. Here, therefore, is a discrepancy which it is necessary 
to get to the bottom of. 

In Peclefs ' Treatise on Heat ' an investigation is given of 
the proper dimensions of a chimney, which investigation is 
recapitulated and ably expanded by Mr. Ranldne. But it gives 
results similar to those deduced from Boulton and Watt's rule 
for their small land boilers, and the expressions are much more 
complicated. Thus if w = the weight of fuel burned in a given 
furnace per second; Y c =the volume of air at 32° required 
per lb. of fuel, and which in the case of common boilers with a 
chimney draught is estimated at 300 cubic feet ; Ti = the abso- 
lute temperature of the smoke discharged by the chimney, and 
which is equal to the temperature shown by the thermometer -f 
461*2°; T =the absolute temperature of the freezing-point, or 
461'2°+ 32° ; A = the sectional area of the chimney in square 
feet ; and u = the velocity of the current in the chimney in feet 
per second: 

Then u— s^ x 



TELOCITY OF DRAUGHT IN CHIMNEYS. 307 

If now I = the length of the chimney and of the flue leading 
to it in feet ; m = the mean hydraulic depth of the smoke, or 
the area of the flue divided by its perimeter, and which for a 
round flue and chimney is \ of the diameter; f= a coefficient 
of friction, the value of which for a current of gas moving over 
sooty surfaces Peclet estimates at 0*012 ; G a factor of resistance 
for the passage of the air through the grate, and which in the 
case of furnaces burning 20 to 24 lbs. of coal per hour on each 
square foot, Peclet found to be 12 ; 7i = the height of the chim- 
ney in feet : Then by a formula of Peclet's 



2 a \ m/ 



9 

which formula, with the value that Peclet assigns to the con- 
stants, becomes 



u* ( 0-012 l\ 
and by transposition and reduction 



, / 64fr & 

v m 

where 64£ is twice the power of gravity, or 32£. 

If now the chimney be made 46 feet high and the flue leading 
to it be 3 feet diameter and 54 feet long, then 64'3 x 46 = 
2957*8 ; -012 x 100 = 1'2 ; m = £th of 3, or f, or "75, and 1*2 4- 
•75 = 1*6. 

Hence the equation becomes 

3 =14-23 
r 14-6 

But u = -j-^ ■ 

Hence w Y ° Tl = 14-23 
AT 

o 

Now if 1,120 lbs. of coal be consumed per hour, *31 lbs. 
will be consumed per second = to ; and if the temperature of 




308 PROPORTIONS OF STEAM-BOILERS. 

the chimney be 600°, then 600° + 461° = 1061° == T*, and 
461° + 32° = 493" = T . 

•31 x 300 x 10G1 
Hence 493A = 14: ' 5 

A = 14 square feet, or 2,016 square inches ; whereas 1,920 
square inches is the area given by Boulton and Watt's rule. 
Peclet's rule, consequently, gives areas much too great for boilers 
with internal flues, though it will answer pretty well for small 
land boilers with external flues : but even here it has the disad- 
vantage of being too complicated for common use. It is clear 
that the friction of the smoke passing through internal flues must 
be much less than the friction of smoke passing through external 
flues like that which surrounds a wagon-boiler. For as only 
one side of the external flues is efficient in heating, the flue with 
the same friction per foot in length Avill require to be nearly 
three times as long as in the case of an internal flue of the same 
area, to give the required amount of heating surface. In steam 
vessels much heat is wasted, from the height of the chimney 
being necessarily so limited that but a small portion of the as- 
censional force due to the temperature of the smoke is obtained. 
Thus, if a height of chimney of 129 feet will produce an exhaus- 
tion of an inch of water when the heat is sufficient to expand 
the air into twice its volume, as will be the case at a tempera- 
cure considerably under 600°, then it is clear that another height 
of 129 feet, added to the first, would produce an exhaustion 
equal to a column of two inches of water without any additional 
expenditure of heat; and this increase would go on until the 
velocity of the draught became such that the friction of the ad- 
ditional height balanced its ascensional force. In steam-vessels, 
where the chimney is necessarily short, a great part of the ex- 
hausting or rarefying effect of the heat is lost ; and in steam- 
vessels, therefore, a chimney-draught is a more wasteful expe- 
dient for promoting combustion than it is in the case of a land 
boiler, where a much larger proportion of the ascensional power 
of the heat may be made available. 

The proportion of heating euffaee per nominal horse-power 



PROPER AREA OP HEATING SURFACE. 309 

obtaining :n marine boilers varies very muoli in different exam- 
ples, being in some boilers 12 square feet, in others 17 square 
feet, in others 20 square feet, in others 80 square feet, and in 
some as much as 85 square feet per nominal horse power. In 
fact, the proportion of heating surface required will depend upon 
the intended ratio in which the nominal is to exceed the actual 
power, which is now often as much, as 8 or 9 times, and also 
upon the measure of expansive action which is proposed to be 
adopted. In marine boilers, as in land boilers, about 9 square 
feet, or 1 square yard, of heating surface will be required to boil 
off a cubic foot of water in the hour, and in Boulton and Watt's 
modern marine tubular-boilers they allow 10 square feet of heat- 
ing surface to evaporate a cubic foot of water in the hour, 10 
square inches of sectional area of tubes, T square inches of sec- 
tional area of chimney, and 14 square inches of area over the 
furnace bridges. The proportions of modern flue-boilers are not 
very^different, except that there is. greater sectional area of flue. 
But no attempt has yet been made to connect the proportions 
proper for small land boilers, with those proper for large marine 
boilers, or to construct a rule that would be applicable to every 
class of flue-boilers. 

Great confusion lias been caused by referring to so indefinite 
a unit as the nominal power of a boiler, and it is much bet- 
ter to make the number of cubic feet which the boiler can 
evaporate the measure of its power. This again depends upon 
the intensity of the draught. But it may be reckoned that 5 or 
6 square feet of surface will evaporate a cubic foot per hour in 
locomotive boilers, and 9 or 10 square feet in land and marine 
boilers. 

The main dimensions and proportions of Boulton and Watt's 
wagon-boilers of different powers are given in the following 
table; — 



310 PROPORTIONS OF STEAM-BOILERS. 



PBOPOETION OF BOULTON AND WATT'S WAGON BOILEES. 



Horse 
Power. 


Length 

of 
Boiler. 


Breadth 

cf 
Boiler. 


Depth 

of 
Boiler. 


Mean 
Height 
of Flue. 


Breadth 

of 
Flue. 


Sectional 

Area 
of Flue. 


Sectional 

Area 
of Flue 1 
per H. P. 




ft in. 


ft in. 


ft in. 


in. 


in. 


sq. In. 


sq. in. 


2 


4 


3 2 


4 1 


20 


9 


ISO 


90 


8 


5 3 


3 4 


4 4 


21 


9 


189 


63 


4 


6 


3 6 


4 7 


22 


10 


220 


55 


6 


7 


3 9 


5 n 


27 


10 


2T0 


45 


8 


8 


4 


5 6 


31 


12 


372 


44 


10 


9 


4 3 


5 9| 


35 


12 


400 


40 


12 


10 


4 6 


6 


36 


13 


468 


39 


14 


10 


4 9 


6 2* 


39 


18 


507 


36 


16 


11 9 


5 


6 6 


40 


14 


560 


35 


18 


12 8 


5 2 


6 8 


42 


14 


588 


32 


20 


13 6 


5 4 


6 11 


44 


14 


616 


30 


30 


16 


5 6 


7 3 


45 


15 


720 


24 


45 


19 


6 


8 5 


53 


16 


795 


17 



These proportions enable us to establish the following rale, 
which is applicable to flue-boilers of every class : — 

TO DETEEMINE THE PEOPEE SECTIONAL AREA OF THE FLUE IN 

FLUE-BOILERS. 

Rule. — Multiply the square root of the number of pounds of 
coal consumed per hour by the constant number 300, and di- 
vide the product by the square root of the height of the chim- 
ney in feet : the quotient is the proper sectional area of the 
flue in square inches. 

Example 1. — What is the proper sectional area of the flue in 
a flue-boiler burning 100 lbs. of coal per hour, the chimney being 
49 feet high. 



Here V 100 = 10, and 10 x 300 = 3000 ; which divided by 
7 (the square root of 49) = 428 square inches, which is the 
proper area of the flue in this boiler. 

Example 2. — What is the proper sectional area of the flue in 
a flue-boiler burning 30 lbs. of coal per hour, the chimney being 
81 feet high ? 

Here ^30 = 5*48, and 5-48 x 300 = 1644; which divided 
by 9 (the square root of 81) = 183 nearly, which is the propei 
area of the flue in square inches. 



BOULTON AND WATT S PRACTICE. 



311 



Example 3. — "What is the proper area of the flue in a flue- 
boiler burning 1,000 lbs. of coal per hour, and with the chimney 
49 feet hiffb ? 



Here ^1000 = 31*78, which x 300 = 9534, and dividing by 
V (which is the square root of 49), we get 1,362, as the proper 
area of the flue in square inches. This is equivalent to 13*62 
square inches per horse-power. 

It is the universal experience with boilers of every class, that 
.&rge boilers are more economical than small, or, in other words, 
that a given quantity of coal will boil off more water in boilers 
of large power than in boilers of small power. Nevertheless, 
for purposes of classification, it may be convenient to assume 
the efficiencies as equal. 

The proper proportions of flue-boilers from 1 to 100 horses 
power are given in the following Table : — 

PROPER PROPORTIONS OF FLTTE-BOILEES OP DIFFERENT POWERS. 



H?rse Power. 


Pounds of 

Coal 
consumed 
per hour. 


Sectional 

Area of Flue 

in B. & W.'s 

boilers. 


Sectional 
Area of Flue 

bv rule, 
with chim- 
ney 49 feet 
high. 


Sectional 
Area of Flue 

by rule, 
with chim- 
ney 81 feet 
high 


Heating 

Surface 

per H. P. 


Sectional 

Area of Flue 

per square ft. 

of heating 

surface. 




lbs. 


sq. in. 


sq. in, 


sq. in. 


sq. ft. 


sq. in. 


1 


10 




123 


106 






2 


20 


*180 


191 


149 


15 


6-0 


3 


30 


189 


235 


183 


13 


4-8 


4 


40 


220 


270 


210 


11 


5-0 


5 


50 




303 


235 






6 


60 


*270 


331 


258 


10-7 


4-2 


7 


70 




353 


278 






8 


80 


*372 


3S3 


296 


10-2 


4-3 


9 


90 




406 


316 






10 


100 


'466 


428 


333 


10 


4-0 


11 


110 




463 


360 






12 


120 


*468 


469 


365 


9-8 


3-9 


13 


130 




483 


380 






14 


140 


*5oi 


507 


394 


9-8 


36 


15 


150 




524 


408 






16 


160 


'560 


541 


421 


9*7 


3-5 


17 


170 




554 


431 






IS 


ISO 


'588 


575 


446 


9-8 


32 


19 


190 




590 


459 






20 


200 


*616 


606 


471 


10 


3-0 


30 


300 


720 


724 


577 


9-8 


2-4 


45 


450 


795 


909 


707 


9-6 


1-7 


60 


600 




1,049 


818 






75 


750 




1,173 


912 






100 


1,000 


1,300 


1.362 


1,059 


8 


16 



312 PROPORTIONS OF STEAM-BOILERS. 

Mr. "Watt reckoned that in his boilers 8 lbs. of coal would 
evaporate a cubic foot of water in the hour, which is equivalent 
to an actual horse-power in the case of engines working without 
expansion. Good Welsh coal, however, it has been found, will 
evaporate 10 lbs. of water for each pound of coal, which is 
equivalent to 1*6 cubic feet of water, or 1*6 horse's power in the 
case of an engine working without expansion ; and if such a 
measure of expansion be used as will double the efficiency of 
the steam, then 10 lbs. of coal burned in the furnace will gene- 
rate 3*2 actual horses' power. To attain this measure of effi- 
ciency, however, the steam would have to be cut off between 
i and \ of the stroke, and in the best boilers and engines work- 
ing with the usual rates of expansion it will not be safe to 
reckon more than 2 (or at most 2-§) actual horses' power as ob- 
tainable by the evaporation of a cubic foot of water. When, 
therefore, engines work up to five times then* nominal power, as 
they now often do, it can only be done by passing through them 
twice the quantity of steam that answers to their nominal power 
— or, in other words, by making the boilers of twice the propor- 
tionate size, unless where some expedient for producing an ar- 
tificial draught is employed. 

The proper height of chimney where the sectional area 

of the flue is known can easily be deduced from the foregoing 

rule. 

J? x 300 . „ (VP x 300) 

For if A = — -t— then h = r 

yfi A 

which formula put into words is as follows : — 

TO FIND THE PEOPEE HEIGHT OF A CHIMNEY IX FEET WHEN THE 
NUMBEB OF POUNDS OF COAL CONSUMED PEE HOUE AND ALSO 
THE SECTIONAL AREA OF THE FLUE AEE KNOWN. 

Rule. — Multiply the square root of the number of pounds of 
coal consumed per hour by the constant nuniber 300, and di- 
vide the product by the sectional area of the flue in square 
inches ; the square of the quotient is the proper height of the 
chimney infect 



DIMENSIONS OF CHIMNEYS FOR GIVEN POWERS. 313 

Example 1. — "What is the proper height of the chimney of a 
boiler consuming 100 lbs. of coal per hour, and with a sectional 
area of flue of 428 square inches. 

Here ^100 = 10, and 10 x 300 = 3000, which divided by 
428 = 7, the square of which is 49, which is the proper height 
of the chimney in feet. 

Example 2. — "What is the proper height of the chimney of a 
flue-boiler consuming 100 lbs. of coal per hour, and with a sec- 
tional area of flue of 333 square inches ? 

Here 4/IOO = 10, and 10 x 300 = 3000, which divided by 
333 = 9, the square of which is 81, which is the proper height 
of the chimney in feet. 

In flue-boilers, the sectional area of the chimney will be the 
same as that of the flue of a boiler of half the power. Hence in 
the foregoing Table the proper sectional area of the chimney of 
a 20-horse boiler — the chimney being 49 feet high — will be the 
same as the sectional area of the flue of a 10-horse boiler, name- 
ly 428 square inches, with a height of chimney of 49 feet ; and 
the proper sectional area of the chimney of a 30-horse boiler 
will be the same as that of the flue of a 15-horse boiler, namely, 
524 square inches, with a height of chimney of 49 feet. If the 
chimney be 91 feet high, then the values will become 333 and 
408 square inches respectively. As then the area of the chimney 
should be the same as that of the flue of the boiler of half the 
power, it is needless to give a separate rule for finding the area 
of the chimney, as such rule will be in all respects the same as 
that for finding the proper area of the flue, except that we take 
half the number of pounds of coal burned per hour instead of 
the whole. 

In marine tubular boilers the total capacity or bulk of the 
boiler, exclusive of the chimney, is about 8 cubic feet for each 
cubic foot of water evaporated per hour — divided in the propor- 
tion of 6 *5 cubic feet devoted to the water, furnaces, aud tubes, 
and 1*5 cubic foot occupied as a receptacle or repository for the 
lileam. The common diameter of tube in marine boilers is about 
8 inches, and the length is 28 or 30 times the diameter. In lo- 
somotive toilers the usual diameter of the tubes is 2 inches, and 
14 



314 



PROPORTIONS OF STEAM-BOILERS. 



the length is about 60 times the diameter. The area of the blast 
orifice in locomotives is about yVth of the area of the chimney. 
The fire-bars are commonly |- inch thick, and the air-spaces are 
made 1 inch wide for fast trains. The main dimensions of ma- 
rine and locomotive boilers required for the evaporation of a 
cubic foot of water, are given in the following Table : — 

PROPORTIONS OF MODEEN BOILERS REQUIRED TO EVAPORATE i 
OUBIO FOOT OF WATER PER HOUR. 



Proportion required per Cubic Foot evaporated 
per hour. 


i 
Marine Flue. Marine Tubu- 

i 


Locomotive. 


Square feet of heating surface 


8 
70 
13 

6 

16 48 
16 


9 to 10 

70 

10 

7 

1S-54 

16 


6 
18 
31 
2-4 

48 

62 




Square inches sectional area of flue or tubes 
Square inches sectional area of chminey. .. 
Square feet of heating surface per square 


Pounds of coal or coke consumed on each 



The quantity of coal or coke burned on each square foot of 
fire-grate in the hour to evaporate a cubic foot of water will of 
course very much depend on the goodness of the coal or coke. 
In the above Table the average working result of 8 lbs. of water 
evaporated by 1 lb. of coal, or a cubic foot of water evaporated 
by 7*8 lbs. of coal, is taken. 

The efficiency of a steam vessel is measured by the expendi- 
ture of fuel necessary to transport a given weight at a given 
speed through a given space, and one of the most efficient steam 
vessels of recent construction is the steamer Hansa, built by 
Messrs. Caird & Co., to ply between Bremen and America. In 
this vessel there are two inverted direct-acting engines, with cy- 
linders 80 inches diameter and 3|- feet stroke. There are four 
tubular boilers, with four furnaces in each, containing a total 
grate surface of 350 square feet, and a heating surface of 9,200 
square feet ; besides which there is a superheater, containing a 
heating surface of 2,100 square feet. The steam is of 25 lbs. 
pressure on the square inch, and it is condensed by being dis- 
charged into a vessel traversed by 3,584 brass tubes, 1 inch ex- 
ternal diameter, and 7 feet long. Each tube having 1*75 squaro 



SURFACE FOR GENERATING AND CONDENSING. 315 

feet of cooling surface, the total cooling surface will be 6,272, or 
about two-thirds of the amount of heating surface. The cooling 
water is sent through the tubes by means of two double acting 
pumps, 21 inches diameter and 24 inches stroke, worked, from 
the forward, end of the crank-shaft. It is much better to send 
the water through the tubes than to send the steam through 
them. But standing and hanging bridges of plate-iron should be 
introduced alternately in the chamber traversed by the tubes, so 
as to compel the current of steam to follow a zigzag course ; and 
the steam should be let in at that end of the chamber at which 
the water is taken off, so that the hottest steam may encounter 
the hottest water. It would further be advantageous to inject 
the feed water into a small chamber in the eduction-pipe, so as 
to raise the feed-water to the boiling-point before being sent 
into the boiler ; or the feed-pipe might be coiled in the eduction- 
pipe so as to receive the first part of the heat of the escaping 
steam. A length of 7 feet appears to be rather great for a pipe 
an inch diameter, as the water at the end of it will become so 
hot as to cease to condense any steam, unless the velocity of the 
flow be so great as to involve considerable resistance from fric- 
tion. Short pipes, with an abundant supply of cold water, will 
enable a very moderate amount of refrigerating surface to suffice, 
as plainly appears from Mr. Joule's experiment, already recited. 
If we reckon the engines of the Hansa at 700 horses' power, 
there will be half a square foot of grate-bars per nominal horse- 
power, and 13*1 square feet of heating surface per nominal 
horse-power in the boiler, besides 3 square feet in the super- 
heater, making in all 16*1 square feet of heating surface per 
nominal horse-power, or 32'2 square feet of heating surface per 
square foot of fire-grate. If we take 9 square feet as evapora- 
ting a cubic foot of water per hour, then the total evaporation 
of the boilers in cubic feet will be 9,200 -~ 9 = 1,022 cubic feet 
per hour ; and if we reckon 8 lbs. of coal as necessary to evapo- 
rate a cubic foot, then the consumption of coal per hour will be 
8,176 lbs, or 3*6 tons per hour, supposing the boiler to be work- 
ing at it-5 greatest power. This is 11*6 lbs. of coal per nominal 
horse-power, reckoning the power at 700 ; and at this rate of 



316 PROPORTIONS OF STEAM-BOILERS. 

consumption 23'2 lbs. of coal will be burned every hour on each 
square foot of fire-grate, to generate the steam rqeuired for a 
nominal horse-power, or it will be 16 lbs. on each square foot 
every hour to evaporate a cubic foot — there heing nearly 1*5 
cubic feet of water evaporated for the production of each nomi- 
nal horse-power. 

INDICATIONS TO EE FULFILLED IX MAELXE BOILEES. 

In all boilers the expedients for maintaining a proper circu- 
lation of the water, so that the flame may act upon solid water, 
and not upon a mixture of water and steam, have been greatly 
neglected ; and the consequence is that a much larger amount of 
surface is required than would otherwise be necessary. The 
metal of the boiler is often bent and buckled by being overheated, 
and priming takes place to an inconvenient extent. In all tubu- 
lar boilers the water should be within the tubes, and those tabes 
should be vertical, so as to enable the cm-rent of steam and water 
to rise upward as rapidly as possible. The best form of steam- 
boat boiler hitherto introduced is the haystack boiler, for which 
we are indebted to the fertile ingenuity of Mr. David Napier, 
and in which boiler the prescribed indications are well fulfilled. 
In the haystack boiler, which is much used in the smaller class 
of river-boats on the Clyde — but which, like the oscillating en- 
gine at the earlier period of its history, has not yet been employed 
in seagoing vessels — the tubes are vertical, with the water within 
them ; and the smoke on its way to the chimney imparts its heat 
to the water by impinging upon the outsides of the tubes. The 
late Lord Dundonald (another remarkable mechanical genius) 
proposed a similar plan of boiler; and boilers on his principle 
— in which the furnace flue of a common marine flue-boiler is 
filled with a grove of small vertical tubes on which the smoke 
impinges on its way to the chimney — have been much used on 
the Continent with good results, and were also introduced in the 
Collins line of steamers navigating the Atlantic. The Clyde 
haystack boilers are generally made of the form of an upright 
cylinder with a hemispherical top, from the centre of which the 
chimney ascends. The furnace is circular, with a water -space 



natier's and dundonald's boilers. 317 

all around it, and with a circular crown ; so that the furnace 
forms, in fact, a short cylinder, divided in some eases into four 
quarters by vertical water-spaces crossing one another. Suitable 
passages are provided to conduct the smoke from the furnace 
into a cylindrical chamber situated above it — the diameter of 
this cylinder being the same as that of the shell of the boiler, 
less the breadth of a water-space which runs round it; and the 
height of this cylinder being equal to the length of the tubes. 
The tubes are set in circles round the chimney ; and the smoke, 
which is delivered from the furnace near the exterior of the 
cylindrical chamber, has to make its way among the vertical 
tubes before it can reach the chimney. The lower tube-plate 
and the furnace crown are stayed to one another by frequent 
bolts, and the cylindrical chamber containing the tubes is also 
bolted at intervals to the shell of the boiler. The water-space 
intervening between the lower tube-plate and furnace crown is 
made very wide, so as to hold a large body of water, and also to 
enable a person to reach in should repairs be required. The only 
weak part of this boiler is the root of the chimney, which some- 
times has collapsed from becoming overheated by the flame as- 
cending the chimney before the steam has been generated ; and 
the small pressure of the air shut within the boiler when heated 
has caused the root of the chimney to collapse. This risk is 
easily prevented by placing several rings of T-iron around the 
root of the chimney, within the steam-chest, and also by carry- 
ing down the plating of the chimney for some distance into the 
tube-chamber, so as to constitute a hanging-bridge that would 
hinder the hottest part of the smoke from escaping, and retain it • 
in the tube-chamber, until it had given out the principal part of 
its heat to the water. In all boilei-3 of this construction these 
precautions should be adopted ; and it would further be useful 
to place a short piece of pipe in the mouth of every upright tube, 
so as to continue the tube up to the water-level, whereby the 
column being elongated its ascensional force would be increased, 
and the circulation of the water be rendered more rapid. 

As this species of boiler is likely to come into use both for 
steam-vessels and for locomotives, it will be proper to indicate 



318 PROPORTIONS OF STEAM-BOILERS. 

the forms which appear to be most suitable for those objects. In 
steam-vessels it is desirable to combine the introduction of a spe- 
cies of boiler adapted for working at a higher pressure, with 
arrangements for burning the smoke, which will be best done 
by maintaining a high temperature in the furnace ; and a high 
degree of heat will be best kept up in the furnace by forming it 
of firebrick instead of surrounding it with water in the usual 
manner. If, therefore, a square box of iron be taken and lined 
with firebrick, and if it be divided longitudinally and transversely 
by these brick walls, and afterwards be arched over, we shall 
have four furnaces, requiring merely the introduction of the fire- 
bars to enable them to be put into operation. Suppose that on the 
top of each of these square boxes a barrel of vertical tubes is 
placed, the barrel being sufficiently sunk into the brickwork to 
establish a communication for the smoke between a hole at each 
of the four top corners of the box and corresponding perforations 
in the barrel, we shall then have the smoke from each of the 
four furnaces into which the box is divided escaping from one 
corner into the chamber containing the tubes, and after travelling 
among them passing to the chimney. In such a boiler the circu- 
lation of the water could be maintained by forming the external 
w ater-space very thick, and by placing a diaphragm-plate in it ; so 
that the water and steam could rise upward on the side of the 
water-space next to the tube chamber, while the solid water de- 
scended on that side of the water-space next to the boiler-shell. 
The intervening plate would enable these currents to flow in 
opposite directions without interfering with one another. 

In a boiler of this kind the grate-bars should have a sufficient 
declivity to enable the coal to advance itself spontaneously upon 
them ; and if there are two lengths of firebars in the furnace, the 
front length should be set closer together than the others, so as 
partially to coke the coal as on a dead-plate, before it enters into 
combustion. This coking would be affected by the radiant heat 
of the furnace, to which heat the coal would be exposed. The 
openings through which the smoke would escape to the tube* 
chamber might be perforations or lattice openings in the brick- 
work, so as to bring every particle of the smoke into intimatG 



IMPORTANCE OF RAPID CIRCULATION. 319 

contact with the incandescent material of which the furnace is 
composed; and these perforations should not have too much 
area, else the heat would escape to the tubes too rapidly, and 
the temperature of the furnace would fall. To maintain a suffi- 
cient draught to bring in the requisite supply of air to the fuel, 
a jet-pipe of steam could be introduced at the bottom of the 
chimney; which jet-pipe would open into a short piece of pipe 
of larger diameter, also pointing up the chimney, and it into 
another larger piece, and so on. The jet at each of these short 
pieces of pipe would draw in smoke and form with the previous 
jet a new jet, which would become of larger and larger volume 
and less velocity at successive steps, until the dimensions of the jet 
had enlarged to an area perhaps equal to half the area of the 
chimney. It will be sufficient if the length of each piece of pipe 
be a little greater than its diameter ; and the lower end of each 
piece, or that end facing the current of smoke, should be opened 
a little into a funnel shape, the better to catch the smoke and 
carry it forward, to form with the steam a jet continually en- 
larging its dimensions. By this mode of construction a powerful 
draught will be created by the jet with a very small expenditure 
of steam. The area through the cylindrical hanging-bridge at 
the root of the chimney should not be large, and the bridge itself 
should be perforated with holes in some places, so as to establish 
a sufficient current of the smoke upward among the tubes to pre- 
vent the heat and flame being swept past direct to the bottom of 
the chimney without rising among the tubes to impart its heat 
to them. 

In the case of locomotive boilers formed with upright tubes, 
the fire-box would be the same as at present; but that part of 
the boiler called the barrel, and which is now filled with longi- 
tudinal tubes, would be formed with flat sides and bottom and a 
semicircular top, so that it would have the same external form 
as the external fire-box, and this vessel would be traversed by a 
square flue, in which the vertical tubes would be set. The sides 
and bottom of this flue would be affixed to the shell by staybolts 
in the same manner as the internal and external fire-boxes are 
stayed to one another; and the top, being semicircular, would 



320 



PROPORTIONS OP STEAM-BOILERS. 



not require staving, while the npper tube-plate forming the top 
of the square internal flue would be strutted asunder and prevent- 
ed from collapsing by the tubes themselves, some of which should 
be screwed into the plates or formed with internal nuts, to make 
them more efficient in this respect. Such a boiler would have 
various advantages over ordinary locomotive boilers, and might 
he made of any power that was desired without any limitation 
being imposed by the width of the gauge of the railway. Such 
boilers might also be used for steam-vessels by merely increasing 
the area of the fire-grate. 

STEEXGTH OF BOELEES. 

The proportions which a boiler should possess in order to havo 
a safe amount of strength will be determined partly by the pres- 
sure of the steam within the boiler, and partly by the dimen- 
sions and configuration of the boiler itself. The best propor- 
tions of the riveted joints of the plates of which boilers are made 
are as follows : — 

BEST PEOPOETIOXS OF EIVETED STEA^I-TIGHT -TOESTS. 











Proper 


Proper 




Proper 


Proper I/ength 


Proper distance 


Quantity of 


Qnan:. 


Thickness of 


Diameter of 


in inches of 


from Centre 


Lap in inches 
in Single 


lap in inches 


Plate in Inches. 


Rivets in 


~..- :-.i :r.~ 


to Centre of 


in Docble 




Inches. 


Head. 


P.ivets in 


Ri-e:c 1 


Riveted 








inches. 


Joints. 


Joists. 


_a. 


3. 


2_ 


H 


H 


2^ 


1 6 


8 


*! 


-■16 


1 
4, 


1 
2 


H 


n 


H 


9-L 


s 
16 


5 

8 


if 


if 


i - 


4 


3. 

8 


4 


H 


if 


2rV 


H 


i_ 


13. 


9i- 


2 


2i 


3f 


1 

8 


1 5 

1 6 


2| 


2h 


2| 


tt 


a 


H 


H 


3 


H 


5/V 



If the strength of the plate iron be taken at 100, then it has 
been found experimentally that the strength of a single-riveted 
joint will be represented by the number 56, and a double riveted 
joint by the number 70. According to the experiments of Messrs. 
Napier and Sons, the average tensile strength of rolled bars of 
Yorkshire iron was found to 61,505 lbs. per square inch of section, 



STRAINS AND STRENGTHS. 321 

and the average strength of bars made by nine different makers 
(and purchased promiscuously in the market) was found to be 
59,276 lbs. per square inch of section. The tensile strength oi 
cast steel bars intended for rivets was found to be 106,950 lbft 
per square inch of section, of homogeneous iron 90,647 lbs., of 
forged bars of puddled steel 71,486 lbs. and of rolled bars of 
puddled steel 70,166 lbs. per square inch of section. The strength 
of Yorkshire plates Messrs. Napier found to be — lengthwise 
55,433 lbs., crosswise 50,462 lbs., and the mean was 52,947 lbs. 
per square inch of section. The tensile strength of ordinary lest 
and lest-best boiler plates, as manufactured by ten different 
makers, was found to be — lengthwise 50,242 lbs., crosswise 45,986 
lbs., and the mean was 48,114 lbs. per square inch of section. 
Plates of puddled steel varied from 85,000 lbs. to 101.000 lbs. 
per square inch of section, and homogeneous iron was found 
to have a tensile strength of about 96,000 lbs. per square inch of 
section. 

Experiments have been made to determine the strength of 
bolts employed to stay the flat surfaces of boilers together ; and 
it has been found that an iron bolt f ths of an inch diameter, like 
the staybolt of a locomotive, screwed into a copper plate f ths 
of an inch thick, and not riveted, bore a strain of 18,260 lbs. 
before it was stripped and drawn out. "When the end of the bolt 
was riveted over it bore 24,140 lbs. before giving way, when the 
head of the rivet was torn off, and the bolt was stripped and 
drawn through the plate. "When the bolt was screwed into an 
iron plate fths of an inch thick, and the head riveted as before, 
it bore a load of 28,760 lbs. before giving way, when the stay was 
torn through the middle. When the staybolt was of copper 
screwed into copper plate and riveted, it broke with a load of 
16,265 lbs., after having first been elongated by the strain one- 
sixth of its length. Locomotive fire-boxes are usually stayed 
with f -inch bolts of iron or copper pitched 4 inches asunder, and 
tapped into the metal of the outer and inner fire-boxes, and the 
stays are generally screwed from end to end. These stays give a 
considerable excess of strength over the shell, but it is necessary 
to provide for the risk of a bad bolt. 
14* 



322 PROPORTIONS OP steam-boilers. 

"With these data it is easy to tell what the scantlings of a 
boiler should be to withstand any given pressure. If we take the 
strength of a single-riveted joint at 34,000 lbs. per square inch, 
then in a cylindrical boiler the bursting strength in pounds will 
be measured by the diameter of the boiler in inches multiplied 
by twice the thickness of the plate in inches, and by the pressure 
of the steam per square inch in pounds ; and this product will be 
34,000 ]bs. Thus in a cylindrical boiler 3 feet or 36 inches diame- 
ter and half an inch thick, if we suppose a length of one inch to be 
cut off the cylinder we shall have a hoop •§• an inch thick and 1 inch 
long. If we suppose one-half of the hoop to be held fast while 
the steam endeavours to burst off the other half, the separation 
will be resisted by two pieces of plate iron 1 inch long and -J an 
inch thick ; or, in other words, the resisting area of metal will be 
one square inch, to tear which asunder requires 34,000 lbs. The 
separating force being the diameter of the boiler in inches mul- 
tiplied by the pressure of the steam on each square inch, and this 
being equal to 34,000 lbs., it follows that if we divide the total 
separating force in pounds by the diameter in inches, we shall 
obtain the pressure of the steam on each square inch that would 
just burst the boiler. Now 34,000 divided by 36 (which is the 
diameter of the boiler in inches) gives 944*4 lbs. as the pressure 
of the steam on each square inch that would burst the boiler. A 
certain proportion of the bursting pressure will be the safe work- 
ing pressure, and Mr. Fairbairn considers that one sixth of the 
bursting pressure will be a safe working pressure ; but in my 
opinion the working pressure should not be greater than between 
one-seventh and one- eighth of the bursting pressure. The 
rule which I gave in my ' Catechism of the Steam Engine,' 
for determining the proper thickness of a single-riveted boiler, 
proceeds on the supposition that the working pressure should be 
r \ of the bursting pressure. That rule is as follows : — 

TO FIND THE PEOPEE THICKNESS OF THE PLATES OF A SINGLE* 
EIVETED OYLINDEICAL P.OILEE. 

Eule. — Multiply the internal diameter of the boiler in inches 
by the pressure of the steam in lbs. per square inch above tht 



STRAINS AND STRENGTHS. 323 

atmosphere, and divide the product by 8,900: the quotient is 
the proper thickness of the plate of the boiler in inches. 

Example 1. — "What is the proper thickness of the plating 
of a single-riveted cylindrical boiler of 3-£ feet diameter, and 
intended to work with a pressure of 80 lbs. on the square 
inch ? 

Here 42 inches (which is the diameter) multiplied by 
80 = 3360, and this divided by 8900 = -377, or a little over f of 
an inch. The decimal '375 is f of an inch. 

Example 2. — What is the proper thickness of a single-riveted 
cylindrical boiler 3 feet diameter, intended to carry a pressure 
of 100 lbs. on the square inch ? 

Here 36 inches x 100 = 3600, which divided by 8900 = '4, 
or, as nearly as possible, -§- and ^ F . 

As the double-riveted joint is stronger than the single-riveted 
hi the proportion of 70 to 56, it follows that 56 square inches of 
sectional area in a double-riveted boiler will be as strong as 70 
square inches in a single-riveted. This relation is expressed by 
the following rule : — 

TO FIND THE PEOPER THICKNESS OF THE PLATES OF A DOUBLE- 
EIVETED CYLINDRICAL BOLLEE. 

Rule. — Multiply the internal diameter of the boiler in incites by 
the pressure of the steam in pounds per square inch above the 
atmosphere, and divide the product by the constant number 
11140 : the quotient will be theproper thickness of the boiler 
in inches when the seams are double-riveted. 

Example 1. — What is the proper thickness of the plates 
of a double-riveted cylindrical boiler 42 inches diameter, and 
intended to work with a pressure of 80 lbs. per square inch ? 

Here 42 x 80 = 3360, and this divided by 11140 = -3016, 
or about ^ of an inch, which is the proper thickness of the 
plates when the boiler is double-riveted. 

Example 2. — What is the proper thickness of a double-riveted 
cylindrical boiler 3 feet diameter, intended to carry a pressure of 
100 lbs. on the square inch ? 



324 PROPORTIONS OF STEAM-B OILERS. 

Here 36 inches x 100 = 3600, which divided by 11140 = -322, 
or a little more than -^ of an inch, which will be the proper 
thickness of the plates of the boiler when the seams are double- 
riveted. 

If T = the thickness of the plate in inches, D = the diameter 
of the cylinder or shell of the boiler in inches,, and P = the 
pressure of the steam per square inch : Then 

D P 

t = oqaTJ is the formula for the thickness of single-riveted 



boilers, and 

DP 



is the formula for double-riveted boilers. 



11140 

Moreover, in single-riveted boilers — 

8900 t n 

d = and 

p 

8900 t 



p = 



D 

So also for double-riveted boilers — 

„ 11140 t -. 
d = and 

p 

11140 t 



p = 

D 

These formulas put into words are as follows : — 

TO FIND THE PEOPEE DIAMETEE OE A SLNGLE-EIVETED BOIIEE 
OF EfOWN THICKNESS OF PLATES AND KNOWN PEESSTJEE OF 
STEAM. 

Rule. — Multiply the thickness in indies by the constant number 
8900, and divide by the pressure of the steam in lbs. per square 
inch. The quotient is the proper diameter of the boiler in 
inches. 

Example 1. — What is the proper diameter of a single-riveted 
cylindrical boiler composed of plates '377 inches thick, and 
intended to work with a pressure of 80 lbs. on the square 
inch? 

Here •377x8900 = 3355-3, which divided by 80=41*94 
inches, or 42 inches nearly, which is the proper diameter in inches. 



STRAINS AND STRENGTHS. 325 

Example 2. — "What is the proper diameter of a single-riveted 
Doiler composed of plates *4 inches thick, and intended to work 
with a pressure of 100 lbs. on the square inch ? 

Here -4 x 8900 = 3560, which divided by 100 = 35'6 inches, 
which is the proper diameter of the cylindrical shell of the boiler 
in this case. 

TO FIND THE PRESSURE TO WHICH A SINGLE-EIVETED CYLINDEIOAL 
BOILEE MAY BE WOEKED WHEN ITS DIAMETEE AND THE THICK- 
NESS OF ITS PLATrNG AEE KNOWN. 

Rule. — Multiply the thickness of the plating in inches by the 

constant number 8900, and divide the product by the diameter 

of the boiler in inches. The quotient is the pressure of steam 
per square inch at which the boiler may be worked. 

Example I. — "What is the highest safe-working pressure in a 
single-riveted boiler 42 inches diameter, and composed of plates 
•377 of an inch thick? 

Here -377 x 8900 = 3355'3, which divided by 42 = 79*8 
lbs. per square inch, which is the highest safe pressure of the 
steam. 

Example 2. — "What is the highest safe-working pressure in the 
case of a single-riveted boiler 36 inches diameter, and composed 
of plates # 4 of an inch thick ? 

Here -4 x 8900 = 3560, which divided by 36 = 99 ibs. per 
square inch. 

The rules for double-riveted boilers are in every case the 
same as those for single-riveted, only that the constant 11140 is 
used instead of the constant 8900. It will therefore be unneces- 
sary to repeat the examples for the case of double-riveted boilers. 

Mr. Fairbairn has given the following table as exhibiting the 
bursting and safe-working loads of single riveted cylindrical 
boilers. But I have already stated that I consider Mr. Fairbairn's 
margin of safety too small. The working pressure, however, 
which he gives for single-riveted boilers would not bo too great 
for double-riveted boilers, as will appear by comparing those 
pressures with the pressures which the foregoing rules indicate 
may bo safely employed. 



S26 



PEOPOETJOISS Of STEAM-BOILERS. 



TABLE SHOWCsG THE BUESTIXG- A5D SAEE-WOEElS'a PBESSTTEE 
OF CYLTNDEICAL BOILEBS, ACOOEDTXG TO ME. EAIEBATEIS". 



1 ' - 

Diameter of 


Working pressure Bursting pressure 


"Working pressure Bnrstin jr pressure 1 


Boiler. 


for |-inch plates. 


for |-mth plates. 


for J^-inca plates. 


for 3^-inch plates. 


ft in. 


lbs. 


lbs. 


Iba. 


lbs. 


3 


118 


70S} 


157} 


944} 


3 3 


109 


658! 


145} 


871! 


S 6 


101 


607 


184J 


809} 


3 9 


941 


566} 


125! 


755} 


4 


ss} 


531 


118 


708} 


4 3 


83} 


500 


111 


666} 


4 6 


78! 


472 


104! 


029} 


4 9 


74* 


447} 


99} 


596} 


5 


7C£ 


425 


94} 


566} 


5 3 


67} 


404£ 


89! 


539} 


5 6 


64! 


886} 


85! 


515 


5 9 


61} 


369} 


82 


492! 


6 


59 


S54 


78! 


472 


6 8 


56} 


340 


75} 


453} 


6 6 


54} 


326f 


72} 


435! 


6 9 


52} 


3143- 


est 


419} 


7 


50} 


303} 


67} 


404} 


7 S 


45J 


293 


65 


396! 


7 6 


47 


288} 


62! 


377} 


7 9 


45} 


274 


60| 


865} 


8 


44 


265| 


59 


354 


8 3 


42* 


257} 


57 


848} 


8 6 


41} 


250 


55} 


888} 


8 9 


4f} 


242f 


54 


828! 


9 


39} 


236 


52} 


314! 


9 6 


37 


220} 


49} 


298} 


10 


i.i- 212} 


47 


25-3} 



It mil be useful to compare some of the figures of this tabic 
with the results given by the roles just recited. For example, 
according to Mr Fairbairn, a single-riveted boiler, 5 feet diame- 
ter, and formed of ^-inch plates, may be habitually worked with 
safety to a pressure of 94J lbs. on the square inch. Now, by our 
rule, '5 x 8900 = 4450, which divided by 60, the diameter of the 
boiler in inches, gives 74 lbs. as the safe pressure at which the 
boiler may be worked. If the boiler be double-rivetdd, then we 
have *o x 11140 = 5570, which, divided by 60, gives 93 lbs. as 
the pressure per square inch at which the boiler may be safety 
worked. This differs very little from Mr. Fairbainrs result of 
94J lbs., and his table may therefore be used if the results be re- 
garded as applicable to double-riveted boilers, but as applied to 
single-riveted boilers his proportions, I consider, are too weak. 
The following diameters of boilers with the corresponding thick- 



COLLAPSING PRESSURE OP FLUES. 327 

ness of plates, it will be seen, are all of equal strengths, their 
bursting pressure being 450 lbs. per square inch, which answers 
to 34,000 lbs. per square inch of section of the iron. Diameter 
3 ft., thickness -250 inches; 3 J ft., -291 ; 4 ft,, -333 ; 4i ft., *376 : 
5 ft., -416; 5$ ft., 458; 6 ft., -500; 6* ft., '541; 7 ft., -583; 7$ 
ft., -625 ; and 8 ft., '666. 

The collapsing pressure of cylindrical flues follows a different 
law from the bursting pressure, being dependent, not merely 
upon the diameter and thickness of the tube, but also upon it3 
length ; and Mr. Fairbairn gives the following formula for com- 
puting the collapsing pressure. If t = the thickness of the iron, 
p = collapsing pressure in lbs. per square inch, l = length of 
tube in feet, and d = diameter of tube in inches ; then 

T 2-19 

p = 806,300 _ 

ID 

and as to multiply the logarithm of any number is equivalent to 
raising the natural number to the power which the logarithm rep- 
resents, we may for t 2 " 19 write 2*19 log. t. "With this trans- 
formation the equation becomes 

p= 806,300 g' 19l0g - T . 

LD 

If now we take the thickness of the plate of the circular flue at 
•291 inches, and if we make the diameter of the flue 12 inches 
and its length 10 feet, the equation will become 

p = 806,300 2-19 log. -291 

' 120 

Now "291 being a number less than unity, the index of its loga- 
rithm will be negative, and for such a number as "291 the index 
will be 1, the minus being for the sake of convenience written 
on the top of the figure; whereas for such a number as ^0291 the 
index will be 2 ; for *00291 the index will be 3, and so on. It 
does not signify, so far as the index is concerned, what the sig- 
nificant figures are, but only at what decimal place they begin ; 
sud -1 has the same index as '291, and -01 as -0291. Now the 
ogarithm of 291, as found in the logarithmicjbables, is 463893, and 
the index being 1, the whole logarithm is 1-468893. In mulli- 



828 



PROPORTIONS OF STEAM-BOILERS 



plying a logarithm with a negative index, as it is the index alone 
that is negative, while the rest of the logarithm is positive, we 
must multiply the quantities separately, and then adding the 
positive and negative quantities together, as we would add a 
debt and a possession, we give the appropriate sign to that 
quantity which preponderates. low '463893 multiplied by 
2-19 = 1-01592567, and T multiplied by 2-19 gives 2M9, which is 
a negative quantity. Adding these products together, we in 
point of fact subtract the 2 # 19 from the 1-01592567, which leaves 
2*82592567. Now if we turn to the logarithmic tables, we shall 
find that the number answering to the logarithm 82592567, or 
the number answering to the nearest logarithm thereto (which is 
825945), is 6698; but as the index is negative, this quantity 
will be a fraction, and the index being 2, the number will begin 
in the second place of -decimals — or, in other words, it will be 
0-6698. Now 806300 multiplied by -06698 = 54004-974, which, 
divided by 120, gives 450 lbs. as the collapsing pressure. If we 
allow the same excess of strength to resist collapse that we 
allowed to resist bursting — namely, 7*6 times— a tube of the 
dimensions we have supposed will be safe in working at a pres- 
sure of 60 lbs. on the square inch. But the strength of tubes to 
resist collapse may easily be increased by encircling them with 
rings of T iron riveted to the tube. Cylindrical flues of different 
dimensions, but of equal strength to resist collapse, are specified 
in the following table : — 



CYLLNDEIOAL FLUES OF EQUIVALENT STEENGTH, THE COLLAPSING 
PEESSUEE BEING 450 POUNDS PEE SQUAEE INCH. 



1 
i 


Thickness of plates in decimal parte of an inch. 


Diameter of 
Flue 










in inches. 


For a Fine 10 feet 


For a Flue 20 feet 


For a Flue SO feet 




long. 


long. 


long. 


12 


•291 


•399 


•4S0 


•18 


•350 


•4S0 


•578 


24 


•399 


•548 


•659 


30 


•442 


•607 


•730 


36 


•480 


•659 


•794 


42 


•516 


•707 


•851 


1 " 


•548 


•752 


•905 



AS ATPLICABLE TO LOCOMOTIVES. 329 

T ' 19 

If p = 806300 , then bv transformation 

T 2-i9 = Zl.?_ and 

806300 

_ 2 ' 19 / PLD 

T_ V 806300* 

If now we put p the collapsing pressure = 450 lbs., l = 10 
feet, and d = 12 inches, the expression becomes 

™ / 54000 loff. -06734 



V t 



806300 2-19 

In like manner the quantities l and d can easily be derived 
from the formula, and in fact the equations representing them 
will be 

806300 t2-19 
l = and 

PD 

806300 T/2-19 



P L 

It is unnecessary to put these equations into words, as the 
rule for finding the collapsing pressure of flues is not much re- 
quired, seeing that in the case of all large internal flues they maj 
be strengthened by hoops of T iron, so as to be as strong as the 
shell. 

PEACTICAL EXAMPLE OF A LOCOMOTIVE F.OILEE. 

It will be useful to compare the results given by these com- 
putations with the actual proportions of a locomotive boiler of 
good construction, and I shall select as the example one of the 
outside-cylinder tank engines constructed by Messrs. Sharp and 
Co. for the ISTorth-Western Eailway. The diameter of cylinder in 
this locomotive is 15 inches, and the length of the stroke 20 inches, 
The pressure of the steam in the boiler is 80 lbs. per square inch. 
The barrel of the boiler is 3 feet 6 inches diameter, and 10 feet 
3£- inches long, and it is formed of iron plates fths thick. The 
junction of the plates is effected by a riveted jump-joint, which 
is equal in strength to a single riveted-joint. The rivets are $ 



330 PROPORTIONS OP STEAM-BOILERS, 

inch in diameter. The external fire-box is of iron fths thick 
and the internal fire-box is -^-ths thick, except the part of the 
tube-plate where the tubes pass through, which is f inch thick. 
The internal and external fire-boxes are stayed together by means 
of copper stay-bolts, ■£ inch in diameter, and pitched 4 inches 
apart. The roof of the fire-box is supported by means of seven 
wrought-iron ribs 1£ inches thick and 3f inches deep, which rest 
el the ends on the sides of the fire-box, while the fire-box crown, 
being bolted to the ribs, is kept up. The ribs are widened out 
at the bolt-holes, and are also made somewhat deeper there, so 
that only a surface of about •§• inch round each bolt bears on the 
boiler crown, to which it is fitted steam-tight. To assist in keep- 
ing up the crown, the cross-ribs are also connected with the 
roof of the external fire-box. The water space left between the 
outside and inside fire-box is about 3 inches, and the inside fire- 
box should always be made pyramidical, to facilitate the disen- 
gagement of the steam from the surface of the metal. There is 
a glass tube and three gauge-cocks, for ascertaining the level of 
the water in the boiler. The lowest gauge-cock is set 3 inches 
above the roof of the internal fire-box, the next 3 inches above 
that, and the next 3 inches above that, so that the highest cock 
is 9 inches above the top of the internal fire-box. 

There is a lead plug £ths of an inch diameter screwed into 
the top of the fire-box. But the usual course now is to place the 
lead plug in a cupped brass plug rising a little way above the 
furnace crown, so that the lead may melt before the plating of 
the crown gets red-hot, should the supply of water be from any 
cause intercepted. 

The boiler is fitted with 159 brass tubes, 10 feet Yf inches 
long, If inches external diameter, and -^th of an inch thick, 
fixed in with ferules only at the fire-box end. Such tubes last 
from four to five years, and they are now made thickest at the 
fire-box end, where the wear is greatest. The part of the boiler 
above the tubes is supported by eight longitudinal stays, running 
from end to end of the boiler. The back tube-plate is of iron 
fjihs of an inch thick. The smoke-box is J inch thick, and the 
chimney, which is 15 inches diameter at bottom and 12^ inches 



AS APPLICABLE TO LOCOMOTIVES. 331 

at top, and rises 13 feet 3 inches above the rails, is |£h of an inch 
thick. The damper for regulating the draught is placed at the 
front of the ash-pan, and there is another similar damper at the 
back of the ash-pan to be used when the engine is made to travel 
backward, which tank engines can the better do, as they have 
no tender. The surface of the fire-grate is 10-fths square feet. 
The steam ports for admitting ihe steam to the cylinder are 11 
inches by If ths, and consequently each has an area of 17*875 square 
inches. The branch steampipe leading to each cylinder has £ 
less area than this. The blast-pipe is 6-| inches diameter, taper- 
ing to 5£ inches diameter at the top, and within it is a movable 
piece of taper pipe, which may be raised up when it is desired 
to contract the blast orifice. The consumption of coke in these 
engines is 25 lbs. per mile. The evaporation in locomotive 
boilers is 7i to 8 lbs. of water per lb. of coke, and in locomotive 
boilers working without expansion the evaporation of a cubic 
foot of water in the hour will be about equivalent to an actual 
horse-power. Now if the speed be supposed to be 30 miles an 
hour, a mile will be performed in two minutes; and as the con- 
sumption per two minutes is 25 lbs., the consumption per one 
minute will be the half of 25 lbs., or say 12 lbs. per minute ; and 
the consumption in 60 minutes, or one hour, will be conse- 
quently 720 lbs. of coke ; and if 8 lbs. of water are evaporated 
by 1 lb. of coke, the water evaporated per hour will be 8 times 
720, or 5760 lbs. !Now if we take a cubic foot of water at 
6&§ lbs., and as the evaporation of a cubic foot in the hour is 
equivalent to a horse-power, 5760 divided by 62^ = 92, will be 
the number of actual horse-power exerted by this engine under 
the circumstances supposed. 

Practically, however, locomotives of this class are capable 
of exerting much more than 92 actual horse-power; for all 
modern locomotives work, to a certain extent, expansively, 
whereby a given bulk of water raised into steam is enabled to 
exert more power, and further, the consumption of coal per 
mile may be increased beyond 25 lbs., with a corresponding in- 
crease of the power generated. In all boilers, indeed, whether 
land, marine, or locomotive, the evaporative power will bo 



662 PROPORTIONS OF STEAM-BOILERS. 

greatly increased by every expedient which increases the velocity 
of the draft, and if arrangements be simultaneously made for in- 
creasing the temperature of the furnace, by contracting the 
escaping orifice over the bridge or through the flues, the expen- 
diture of fuel to accomplish any given evaporation will not be 
increased. In this way marine boilers have been constructed 
with only 12 square feet of heating surface per nominal horse 
power, and in which the consumption was only 2 J- lbs. of coal 
per actual horse power, as will be seen by a reference to page 52 
of the Introduction to my - Catechism of the Steam Engine.' 



CHAPTER VI. 

i*OWER AND PERFORMANCE OF ENGINES. 

Tk& manner of determining the nominal power of an engine 
Las been already explained, and it now remains to show in what 
manner its actnal or indicator horse-power may be determined. 

Construction of the Indicator. — The common form of indica- 
tor applicable to engines moving at low rates of speed I have al- 
ready described in my ' Catechism of the Steam-Engine.' But 
in the case of engines moving at high rates of speed, and, in fact, 
in the case of all engines to which the steam is quickly admitted, 
the diagrams formed by this species of indicator are much dis- 
torted, and the accuracy of the result impaired, by the momen- 
tum of the piston of the indicator itself, which is shot up sud- 
denly by the steam to a point considerably higher than what 
answers to the actual pressure. The recoil of the spring again 
sends the piston below the point which properly represents the 
pressure ; and in interpreting the diagram the true curve is sup- 
posed to run midway between the crests and hollows of the 
waving line produced by these oscillations. Latterly an im- 
proved form of indicator, called Eichards' indicator, has been 
introduced, which is represented in fig. 5, of which the main pe- 
culiarity is that its piston is very light and has a very small 
amount of motion, so that its momentum is not sufficiently great 
to disturb the natural line of the diagram. The motion of the 
piston of the indicator is multiplied sufficiently to give a diagram 
of the usual height by means of a small lever jointed to the top 
of the piston rod. To the end of this lever a small link, carry- 



334 



EICKASDS' IKDICATOS. 

Fig. 5. 





eiceaiids' irr.iCATOit. (By Elliot Brothers, Strand.) 



METHOD OF APPLYING THE INDICATOR. 335 

ing the pencil, is attached, and from the lower end of this small 
link a small steel radius bar proceeds to a fixed centre on a suit- 
able part of the instrument, so as to form a parallel motion 
whereby the pencil is constrained to move up or down in a ver- 
tical direction. The paper is placed upon the drum, shown in 
the figure with a graduated scale, and the string causing this 
drum to turn round and back again on its axis is put into con- 
nection with some part partaking of the motion of the piston in 
the usual manner. To withdraw the pencil from the paper, the 
whole parallel motion and the arms carrying it are turned round 
upon the cylinder, and the pencil is thus made readily accessible. 
The action of this indicator is precisely the same as that of the 
common indicator, which, having been described in my ' Cate- 
chism of the Steam-Engine,' need not be further noticed here. 
But in this indicator, as the spring is very stiff, and the travel of 
the piston correspondingly small, there are no inconvenient os- 
cillations of the pencil such as occur when a long and slender 
spring is employed. 

Method of applying the Indicator. — The drum being put into 
communication with some part of the engine possessing the same 
motion as the piston, but sufficiently reduced in amount to be 
suitable for the small size of the instrument, the drum will begin 
to be turned round when the piston begins its forward stroke ; 
and the string having drawn it round in opposition to the ten- 
sion of the spring coiled at the bottom of it, it will follow that 
when the string is relaxed, as it will be on the return stroke of 
the piston, the drum will turn back again to its original position, 
and its motion and that of the string will be an exact miniature 
of the motion of the piston. The pencil, if now suffered to press 
against the paper, will describe a straight line. But if the cock 
which connects the cylinder of the indicator with the cylinder 
of the engine be now opened, the pencil will no longer trace a 
straight line, but being pressed upward during the forward 
stroke by the steam, and being sucked downward by the vacuum 
during the return stroke, if the engine is a condensing one, or 
being pressed downward by the spring when the pressure of tho 
steam is withdrawn, as it will be during the return stroke, it is 



336 



POWER AND PERFORMANCE OF ENGINES 



quite clear that the pencil must now describe a figure containing 
a space or area, and the figure is what is called the indicator di- 
agram, and the amount of the space is the measure of the amount 
of the power exerted at each stroke by the engine. This will be 
more clearly understood by a reference to fig. 6, which is an in- 
dicator diagram taken from a steam fire-engine constructed by 
Messrs Shand, Mason and Co., with two high-pressure engines 
of 6£ inch cylinders and 7 inches stroke, with a pressure on the 

Fig. 6. 



Eduction 
corner. 




S (<W(i Forward stroke. 
NXA * Steam line. 


Admission 
corner. 


tSs. 

140- 
j 130- 
















> 


1 120- 

! no— 

! ioo- 

j 90- 
i 90- , 


'/ ° 

!l — 


o, 

fcs 


o 

o 
£2 


o 


121 «5 
121 • 


o 


o ! \r> 

— I CM 


O 
O 


i 7 "1 

j 60-1 
! 50- 

i 40- 
















\ 30 " 
















/ i 2 °- 










EDUClTION 


LIME 




/ 10- 

















Atmospheric line 
Return stroke. ^> > 



Compressive 
corner. 



DIAGRAM ILLUSTRATIVE OF THE MODE OF COMPUTING THE HORSE-POWER. 

boiler of 145 lbs. per square inch, and making 156 revolutions 
per minute. The total weight of this engine is 24 cwt. 2 qrs., 
and by a reference to the diagram it will be seen that the mean 
pressure urging the piston is 117'5 lbs. per square inch, which 
mean pressure is ascertained by adding together the pressure at 
each division or ordinate, and dividing by the number of ordi- 
nates, which in this case is 1 0. The mean pressure multiplied 
by the areas of the cylinders and by the speed of the piston in 
feet per minute, and divided by 33000 lbs., gives 18'3 horses aa 



INTERPRETATION OP INDICATOR DIAGRAMS. 337 

the power actually exerted by this engine. The weight of the 
engine is consequently only 1*3 cwt. per actual horse-power. 

The advantage of taking 10 ordinates instead of 8 or 9 or 11 
is, that the division by 10 is accomplished by merely shifting the 
position of the decimal point; while 10 ordinates are enough to 
enable the area to be measured accurately enough for all practi- 
cal purposes. Thus the total amount of the pressures in the di- 
agram, fig. 6, taken at 10 places, is 1175 lbs., and the tenth of 
this, or 117*5 lbs. per square inch, is the mean pressure on the 
piston throughout the stroke. It is clear that when we have 
got the mean pressure on each square inch of the piston, we 
have only to ascertain the number of square inches in it, and 
the distance through which it moves in a minute, to determine 
the power, and the indicator enables us to determine the mean 
pressure on the piston throughout the stroke in the manner just 
explained. The indicator is sometimes applied to the air-pump 
and to the hot well, to determine the varying pressures within 
them at different parts of the stroke; and it is virtually the 
stethoscope of the engine, as it enables us to tell whether all its 
internal motions and pulsations are properly performed. 

Mode of reading Indicator Diagrams. — In the preceding di- 
agram the piston moves in the forward stroke in the direction 
shown by the arrow, and backward on the return stroke in the di- 
rection shown by the arrow. In all diagrams the top indicates the 
highest pressure, and the bottom the lowest pressure. But it is 
quite indifferent whether the diagram is a right-hand or left- 
hand diagram ; and where two diagrams are shown on the same 
piece of paper, as is often done, that which represents the per- 
formance of one end of the cylinder is generally right-hand, and 
that'which represents the performance of the other end of the 
cylinder is generally left-hand. This arrangement, however, is 
quite immaterial, that which alone determines the power exert- 
ed being with any given scale the area shut within the diagram. 

In fig. 6, the steam being supposed to be let in upon the pis* 

ton of the engine, presses the piston of the indicator up to the 

point shown at the 'admission corner,' and as the piston moves 

forward the steam continues to press upon it with undiminished 

15 



338 POWER AND PEEFORltANCE OF ENGINES. 

pressure, until close to the end of the stroke, at the ' eduction 
corner,' the eduction passage is opened ; and as the steam con- 
sequently escapes into the atmosphere there is no longer the 
same pressure on the spring of the indicator as before, and its 
piston consequently descends. As, however, the steam cannot 
instantaneously get away, the pressure does not descend quite so 
low as the atmospheric line. The eduction passage, it appears 
by the diagram, begins to be opened when about nine-tenths of 
the forward stroke has been completed, and it also begins to be 
shut when about nine-tenths of the return stroke has been ccm- 

Rff. 7. 




DIAGRAM IAKKS FROM STEAMER ' ISLAXD QUEEN.' 



pleted, as appears by a reference to the ' compression ' corner, 
which shows that the back pressure begins to rise before the 
termination of the stroke. The area comprehended between the 
atmospheric line and the bottom of the diagram shows the 
amount of back pressure resisting the piston, which in this dia- 
gram is of the average amount of 5-1 lbs.; and this increased 
back pressure at the 'compression corner' is produced by the 
compression of the steam shut within the cylinder, which is ac- 
complished by the piston as it approaches the end of its stroke. 
Various examples of Indicator Diagrams.— -In the engine of 
which the diagram is given in fig. 6, the steam works with very 
little expansion ; but in fig. *l we have a diagram taken from the 
steamer 'Island Queen which shows a large amount of expan- 



" INTERPRETATION OF INDICATOR DIAGRAMS. 339 

Biou. This diagram is a left-hand diagram, the former one, 
shown in fig. 6, being a right-hand diagram, a is the admission 
corner, and the steam is only admitted until the piston reaches 
the position answering to that of a vertical line drawn through 
#, and which is about one-eighth of the stroke. The steam be- 
ing shut off from the cylinder at a, thereafter expands until the 
end of the stroke is nearly reached, when the eduction passage 
is opened, and the pencil then subsides to the point b, at which 
point the piston begins to return. The straight line drawn 
across the middle of the diagram is the atmospheric line ; and it 
is traced by the pencil before the cock of the indicator commu- 
nicating with the cylinder is opened. The distance of the line 
b o below the atmospheric line shows the amount of vacuum ob- 
tained in the cylinder, and the height of a a above the atmos- 
pheric line shows the pressure of the steam subsisting in the 
cylinder. This diagram, which is a very good one, is obtained 
with the aid of a separate expansion valve. The pressure of the 
steam was 22 lbs. per square inch, the vacuum 14-J lbs., and the 
number of revolutions per minute 17. 

In some high-pressure engines, where the steam is allowed to 
escape suddenly through large ports, and a large and straight 
pipe, there is not only no back pressure on the piston, but a 
partial vacuum is created within the cylinder by the momen- 
tum of the escaping steam. In ordinary condensing engines 
the momentum of the steam escaping into the condenser might 
in some cases be made to force the feed-water into the boiler, in 
the same manner as is done by a Giffard's injector, which is an 
instrument that forces water into a boiler by means of a jet of 
steam escaping from the same boiler. This instrument will not 
act if the temperature of the feed-water be above 120° Fahr., as 
m such case the steam will not be condensed with the required 
rapidity. As the steam is water in a state of great subdivision, 
and as the particles of this water are moved with the velocity 
of the issuing steam, which is very great, we have in effect a 
very small jet of water issuing with a very great velocity, and 
this small stream would consequently balance a very high head 
of water, or, what comes to the same thing, a very great pres- 



340 



POWER AND PERFORMANCE OF ENGINES. 



sure. Precisely the same action takes place "when the steam es- 
capes to the condenser ; and under suitable arrangements the 
boiler might be fed by aid of the power resident in the educting 
steam, and iudeed the function of the air-pump might also be 
performed bj the same agency. 

In fig. 8 we have an example of the diagrams taken from the 
tcp and the bottom of the cylinder disposed on the same pieco 
of paper, those on the left-hand side being taken from the top 
of the cylinder, and those on the right-hand side being taken 

Fig. 8. 



M-i 




DIAGRAMS TAKEN AT MOORINGS FROM HOLYHEAD PADDLE-STEAMER 
' MDNSTER.' 



from the bottom of the cylinder. There are three diagrams 
taken from each end with different degrees of expansion, a is 
the admission corner of the three diagrams, taken from the top 
of the cylinder, and a a a are the three several points at which 
the steam is cut off m these three diagrams. Thereafter the 
steam continues to expand, and the pressure gradually to fall, 
•until the points 5 5 5 are reached, when the eduction passage is 
opened to the condenser, and the pressure then falls suddenly to 
ihe point b. The liue b b' represents the amount of exhaustion 



DIAGRAMS OF HOLYHEAD STEAMER. 



341 



attained within the cylinder measured downward from the at- 
mospheric line m l ; and ccc represent the three points at which 
compression begins, answering to the three degrees of expansion. 
The letters a', a\ V, b', and c' represent the corresponding points 
for each of the three diagrams taken from the bottom of the cyl- 
inder; and the amount of correspondence in the right-hand and 
left-hand diagrams shows the amount of accuracy with which 
the valves are set to get a similar action at each end of the cyl- 

Fig. 9. 




DIAGKAMS TAKEN FROM HOLYHEAD FADDLE-STEAMER ' ULSTER ' WHEN 

UNDER WAT. 



\nder. The diagrams given above were taken from the Holy- 
head steam-packet 'Munster,' the engines of which were con- 
structed by Messrs. Boulton and Watt. The cylinders are oscil- 
lating, of 96 inches diameter and T feet stroke. The pressure of 
steam was 26*16 lbs. per square inch, vacuum 25| lbs., and the 
number of strokes per minute 9 — the vessel having been at moor- 
ings at the time. It will be seen by these diagrams that the amount 
of lead upon the eduction side, or the equivalent distance which 
the piston is still from the end of the stroke when eduction begins 
to take place, corresponds in every instance with the amount of 



342 



POWER AND PERFORMANCE OF ENGINES. 



the compression, since, in fact, by shifting the eccentric ronnd 
to let the steam out of the cylinder "before the end of the stroke, 
the valve will be equally shifted to shut the educting orifice be- 
fore the end of the stroke, and thus to keep within the cylinder 
any vapour left in it when the valve has been shut, and which 
is thereafter compressed by the piston until the end of the stroke 
is reached, or until the valve opens the communication with the 
boiler. 

Fig. 9 represents a diagram taken from the top, and another 
taken from the bottom of one of the cylinders of the Holyhead 
paddle-steamer ' Ulster,' a vessel of the same power and dimen- 
sions as the 'ilunster,' and the engines also by Messrs. Boulton 

Fiff. 10. 




DIAGRAMS FROM STEAMER ' ULSTER ' AT 4 1-2 STROKES. 
(STEAM THEOTTLED BY THE LESX.) 

and "Watt. "When these diagrams were taken the pressure of the 
steam in the boiler was 28 lbs. per square inch, the vacuum in 
the condenser 13 lbs. per square inch, and the engine was mak- 
ing 23 strokes per minute. The mean pressure on the pistons, 
obtained by taking a number of ordinates, as in fig. 6, reckoning 
up the collective pressure at each, and dividing by the number 
of ordinates, was 28'27 lbs. It is immaterial what number of 
ordinates is taken, except that the more there are taken the more 
accurate will be the result. 

In fig. 10 we have diagrams taken from top and both on in the 
Barne engine, when slowed to 4J strokes per minute, partly by 
olosing the throttle valve, and partly by shifting the link towards 
its mid-position. In these diagrams nearly the whole areas are 



DIAGRAMS OF HOLYHEAD MAIL STEAMER. 343 

below the atmospheric line. But on the left-hand corner of one 
of the figures a loop is formed, which often appears in engines 
employing the link, and the meaning of which it is necessary to 
explain. The extreme point of the diagram in every instance 
answers to the length of the stroke ; and if the steam is pent up 
in the cylinder by the eduction passage being shut before the 
end of the stroke, or if it be suffered to enter from the boiler be- 
fore the stroke is ended, the pencil will be pushed up to its high- 
est point before the stroke is ended, and as the paper still con- 
tinues to move onward the upper part of the loop is formed. If 
the pressure within the cylinder when the piston returns were 
to be precisely the same as when the piston advances during this 
part of its course, the loop would be narrowed to a line. But as 
the advance of the piston when the valve is very little opened 
somewhat compresses the steam, and as its recession when the 
valve is very little opened somewhat wire-draws it, the pressures 
while the piston advances and retires through this small distance, 
although the cylinder is open to the boiler by means of a small 
orifice, will not be precisely the same ; and the higher pressure 
will form the upper part of the loop, and the lower pressure the 
lower part. In fig. 10, by following the outline of the left-hand 
diagram, it will be seen that the steam begins to be compressed 
within the cylinder when about three-fourths of the stroke has 
been completed; and the pencil consequently begins to rise 
somewhat above its lowest point. But as the vapour within the 
cylinder is very rare, the rise is very little until, when the piston 
is about one-eighth part of its motion, or about 8 inches from 
the end of the stroke, the steam-valve is slightly opened, when 
the piston of the indicator is compelled to ascend to the point 
answering to the pressure within the cylinder thus produced. 
As the opening from the boiler continues, and the piston by ad- 
vancing against the steam, instead of receding from it, compress- 
es rather than expands the steam admitted into the cylinder, tha 
pressure continues to rise somewhat to the end of the stroke ; 
when the piston of the engine, having to move in the opposite 
direction, the steam within the cylinder will be expanded, and 
any still entering will be wire-drawn in the contracted passage, 



344 



POWER AND PERFORMANCE OF ENGINES. 



and the pressure will fall. Under such circumstances a loop will 
necessarily be formed at the corner of the diagram, such as is 
shown to exist at the left-hand corner of fig. 10. The reason 
why there is no corresponding loop at the right-hand corner of 
the right-hand diagram is simply because the valve is somewhat 
differently set at one end of the engine from what it is at the 
other ; and the angles of the eccentric rods will generally causa 



Fig. 11. 



7fo20r- 




DIAGRAM FROM AIR-PUMP OF STEAMER ' ULSTER. 
(19 REVOLUTIONS PEE MINUTE.) 

some small difference in the action of the valve at the different 
ends of the engine. 

Diagrams from the Air-Pump. — Fig. 11 is a diagram taken 
from the air-pump of the 'Ulster,' when the engine was making 
19 revolutions per minute. In this diagram the pencil begins to 
ascend from that point which marks the amount of exhaustion 
existing in the air-pump, and it rises very slowly until about 
two-thirds of the stroke of the pump has been performed, when 
it shoots rapidly upwards, indicating that at this point the water 
is encountered which has to be expelled. Midway between tho 



DIAGRAMS TAKEN FROM THE AIR-PUMP. 345 

atmospheric line and the highest point of ascent, the delivery 
valve begins to open, and somewhat relieves the pressure ; and 
there is consequently a wave in the diagram on that point. But 
the inertia of the water in the hot- well has then to be encoun- 
tered, and an amount of pressure is required to overcome this 
inertia, which is measured by the highest point to which the 
pencil ascends. So soon as the water in the hot-well and waste- 
water pipe has been put into motion, the motion is continued 
by its own momentum, without a sustained pressure being re- 
quired to be exerted by the bucket of the pump ; and the pres- 
sure in the pump consequently falls, as is shown by the descent 
of the piston of the indicator towards the end of the stroke. 

Fig. 12. 

lts.5— 




DIAGRAM FROM AIR-PUMP OF STEAMER ' ULSTER.' 
(T STROKES PER MINUTE.) 

The effect of partially closing the throttle-valve of an engine 
so as to diminish the speed, will be to reduce the momentum of 
the water in the hot-well, and correspondingly to reduce the 
maximum pressure which the pump has to exert. But the ef- 
fect will also be to fill the pump with water through a larger 
proportion of its stroke ; and if the engine were to be slowed 
very much by shutting off the steam, without correspondingly 
shutting off the injection, the air-pump at its reduced speed 
would be unable to deliver all the water, which would conse- 
quently overflow into the cylinder and probably break down the 
engine. In fig. 12 we have an air-pump diagram taken from the 
steamer ' Ulster,' when the speed of the engine was reduced to 
15* 



3-16 POWER AND PERFORMANCE OF ENGINES. 

six strokes per minute ; and it "will be observed tbat we have 
no longer the same amount of maximum pressure in the pump, 
nor the same sudden fluctuations. The pump, however, is rilled 
for a greater proportion of its stroke ; and the maximum pres- 
sure once created, is constant, and does not rise much above the 
pressure of the atmosphere, being, in fact, the simple pressure 
due to the pressure of the atmosphere, and that of the column 
of water intervening between the level of the air-pump and that 
of the waste-water pipe. 

Diagram illustrative of the evils of Small Ports. — Fig. 13 is 
a diagram taken from a pumping-engine in the St. Katherine's 
Docks, and is introduced mainly to show the detrimental effect 

Fie. 13. 



DIAGRAM TAKEX FROM PU.MPIXG-EXGINE, ST. KATHERIXE S DOCKS. 

of an insufficient area of the eduction passages. The steam is 
supposed to enter at the left-hand corner, but as the speed of the 
piston accelerates, as it does towards the middle of the stroke, 
the pressure falls, from the port being small and the steam wire- 
drawn. Towards the other end of the stroke the pressure would 
again rise, but that it is hindered from doing so by the condensa- 
tion within the cylinder, which is considerable, as the engine 
works at the low speed of 12 strokes per minute, lifting the wa- 
ter 9-| feet. The eduction corner of the diagram is very much 
rounded away, from the inadequate size of the ports ; and the 
eduction will also be impeded by any condensed water within 
I he cylinder, which, unless got rid of by other arrangements, will 
have to be put into motion by the escaping steam. The mean 



DIAGRAMS TAKEN FROM THE HOT-WELL. 



347 



pressure exerted on the piston of this engine is only 12*45 lbs. 
per square inch, although it operates without expansion ; and it 
may be taken as a fair example of eneligible construction. 

Diagrams showing the momentum of the Indicator piston. — 
Fig. 14 is a pair of diagrams taken from one of the engines of 
H. M. S. ' Orontes.' This vessel, which is 300 feet 1 inch long, 
44 feet 8 inches broad, and 2,823 tons, has horizontal direct 
acting engines of 500 horse-power, constructed by Messrs Boul- 

Fig. 14. 




diagram; taken from h.m. troop-steamer 'orontes.' 

ton and "Watt. "With a midship section of 644 square feet, and a 
displacement of 3,400 tons, the vessel attained a speed on her offi- 
cial trial, of 12*622 knots, with a pressure of steam in the boiler 
of 25 lbs. per square inch, 61 revolutions per minute, the engines 
exerting 2,249 horse-power. On one occasion the speed obtained 
was 13*3 knots. "With an area of immersed section of 781 square 
feet, and a displacement of 4249 tons, the speed attained was 
12*354 knots, with 2,143 horse-power. There are two horizontal 



848 POWER AND PERFORMANCE OP ENGINES. 

engines of 71 inches diameter, and 3 feet stroke. The screw is 
18 feet diameter, 25 feet pitch, and 4 feet long, and the slip of 
the screw was found to vary between 13 and 16 per cent. When 
the diagrams represented in fig. 14 were taken, the pressure of 
the steam in the boiler was 2H lbs. of the vacuum, in the con- 
denser lly lbs., and the engine was making 60 revolutions pei 
minute. If ordinates be taken in the case of these diagrams, and 
the mean pressure be thus determined, it will be found to amount 
to 25*22 lbs. per square inch. In these diagrams the waving line 
formed by the pencil, owing to the momentum of the piston of 
the indicator, is very plainly shown ; and although such irregu- 
larities will not materially impair the accuracy of the result, if a 
sufficient number of ordinates be taken correctly to measure the 
irregularity, yet it is greatly preferable to employ an indicator 
which will be as free as possible from the disturbing influence of 
the momentum of its own moving parts. In this engine, as in 
most of Messrs. Boulton and "Watt's engines, there is a great 
similarity in the diagrams taken from each end of the cylinder 
— a result mainly produced by giving a suitable length to the 
eccentric rods, by moving up or down the links vertically by 
a screw, instead of by a lever moving in the arc of a circle, and 
placing the projecting side of the eccentric suitably with the 
curvature of the link, since, if placed in one position, it will aggra- 
vate the distortion produced by the angle of the eccentric rods, 
and if placed in the opposite position it will correct this dis- 
tortion. 

Fig. 15 represents a series of diagrams from each end of one 
of the engines of the ' Orontes,' formed by allowing the pencil to 
rest on the paper during many revolutions, instead of only dur- 
ing one. These diagrams show small differences between one 
another, mainly in the mean pressure of the steam. 

Fig. 16 represents two diagrams taken from the engines of 
the iron-clad screw steamer 'Eesearch, fitted with horizontal 
engines, with 50-inch cylinders, and 2 feet stroke. With a pres 
sure of steam in the boiler of 22 lbs., and with a vacuum in the 
condenser of 12f lbs. per square inch, the mean pressure on the 
piston shown by the diagrams is 24*55 lbs., the engine making 



DIAGRAMS FROM DIFFERENT STEAMERS. 



349 



85 revolutions per minute. This engine is fitted with surface 
condensers. The serrated deviation at a is caused by the mo- 
mentum of the piston of the indicator. 

In fig. 17 we have two diagrams, taken from opposite ends 
of one of the engines of H.M.S. ' Barossa.' This vessel is 225 
feet long, 40 feet 8 inches broad, and 1,702 tons burden. "With 
a mean dranght of water 15-J feet or thereabout, the area of mid- 
ship section is 466 square feet, and the displacement 1,780 tons. 
The vessel is propelled by two horizontal engines, with cylinders 

Fig. 15. 




DIAGRAMS TAKEN FROM SCREW STEAMER ' ORONTES.' 

of 64 inches diameter and 3 feet stroke, the nominal power 
being 400 horses. On the olncial trial this vessel realised a 
speed of 11 "92 knots, with a pressure of steam in the boiler of 
20 lbs. per square inch, and with an indicated power of 1798*2 
horses, the engine making 66 revolutions per minute. The 
screw is 16 feet diameter, 24 feet pitch, and 3 feet long, and the 
slip at the time of trial was 23*71 per cent. When the diagrams 
shown in fig. 17 were taken, the pressure of steam in the boiler 
was 19 lbs. per square inch ; vacuum in condenser 12 J lbs. per 
square inch, the revolutions 66 per minute, and the mean pres- 



350 



POWER AND PERFORMANCE OF ENGINES. 



sure on the piston 22*3 lbs. per square inch. The area of a cylin- 
der of 64 inches diameter is 3216*2 square inches, the double of 
which (as there are two cylinders) is 6433*8 square inches, and 
as there 22*3 lbs. on each square inch, there will be a total pres- 
sure of 6433*8 times 22*3, or 143,473*74 lbs. urging the pistons, 
and as the length of the double stroke is 6 feet, the power ex- 
erted will be equal to 6 times 143,473*74 lbs., or 860,840*44 

Fig. 16. 






INDICATOR DIAGRAMS FROM IRON-CLAD STEAMER * RESEARCH.' 

foot-pounds per stroke, and as there are 66 strokes per minute, 
there will be 66 times this, or 56,797,869*04 foot-pounds exerted 
per minute. As an actual horse-power is 33,000 foot-pounds 
per minute, we shall, by dividing 56,797,869*04 by 33,000, get 
the actual power exerted by this engine at the time the above 
diagrams were taken, and which, by performing the division, we 
shall find to be 1721*1 horses. 

Various Diagrams. — Fig. 18 is a diagram taken from the 
air-pump of the 'Barossa,* which is a double-acting pump. The 



DIAGRAMS FROM DIFFERENT STEAMERS. 



351 



injection was all on at the time this diagram was taken, and the 
vacuum was only 11 lbs. per square inch. In my ' Catechism 
of the Steam-Engine,' published in 1856, 1 drew attention to the 
fact of the existence of very imperfect vacuums in engines with 

Fig. 17. 



-10 




DIAGRAMS TAKEN FROM H. M. STEAMER 'bAROSSA.' 

double-acting air-pumps, the buckets of which move at a high 
rate of speed ; and I also pointed out the cause of this imperfect 
vacuum, which I showed to be consequent on the lodgment of 



Fig. 18. 



USr- 10 




-5. 



AIR-PUMP DIAGRAM FROM H. M. STEAMER ' BAROSSA.' 

iarge quantities of water between the foot and delivery-valves 
at the end of the pump, into which water the pump forced in 
the air or drew it out without ejecting it from the pump at all. I 
consequently recommend that in all pumps of this class the bucket 



552 POWER AND PERFORMANCE OP ENGINES. 

and valve-chambers should be so contrived that every particle 
of water would be forced out of the pump at every stroke. But 
up to the present time I do not find that this recommendation 
has been generally adopted, and in nearly every species of direct- 
acting screw-engine operating by a jet in the condenser, the 
vacuum is much worse than it was in the old class of paddle- 
engines, or even in the land engines made by Watt nearly a cen- 
tury ago. 

In fig. 19 we have an example of diagrams taken from the top 
and bottom of one of the paddle-engines of the steamer ' Great 
Eastern,' constructed by Messrs. J. Scott Eussell and Co. These 
engines are oscillating engines of 74 inches diameter of cylinder, 
and 14 feet stroke, making 10 revolutions per minute, and there are 

Fig. 19. 




DIAGRAMS FROM PADDLE-EXGIXES OF ' GREAT EASTERN.' 

four cylinders, or two to each wheel. The mean pressure on the 
piston which these diagrams exhibit is 22*2 lbs. per square inch, 
from which, with the other particulars, it is easy to compute the 
power. 

In fig. 20 we have two different pairs of diagrams. The 
larger pair is taken from one of the engines of the paddle- 
steamer ' Ulster,' and the smaller pah* — represented in dotted 
lines — is taken from the engines of the paddle-steamer ' Victoria 
and Albert.' In the case of the ' Ulster ' the pressure of steam 
in the boiler when the diagram was taken was 26 lbs. per square 
inch, and the vacuum in the condenser 13 lbs. per square inch. 
The number of strokes per minute was 23, the mean pressure on 
the piston 28*77 lbs. per square inch, and indicated horse-powei 



DIAGRAMS FROM DIFFERENT STEAMERS. 



353 



4,100. The ' Victoria and Albert * has two oscillating engines, with 
88-inch cylinders and 7-feet stroke. The pressure of the steam 
in the boilers when the diagrams were taken was 26 lbs. per 
square inch ; of the vacuum 12-| lbs. per square inch ; the mean 
pressure on the piston 22*87 lbs. per square inch, and the num- 
ber of strokes per minute 25 '4. The area of an 88-inch cylinder 
is 6082*1 square inches, and the area of two such cylinders is tho 

Ms*. 20. 




COMPARATIVE DIAGRAMS FROM '■ ULSTER ' AND ' VICTORIA AND ALBERT.* 



double of this, or 12,164*2 square inches, and as there are 22*87 
lbs. on each square inch, the total pressure urging both pistons 
will be 12,164*2 times 22*87 or 278,195 lbs. Now, as the 
length of the stroke is 7 feet, and as the piston traverses it each 
way in each revolution, the piston will travel 14 feet for each 
revolution, and 278,195 multiplied by 14 will give 3,894,730 as 
the number of foot-pounds exerted in each stroke ; or, as there 
are 25*4 strokes each minute, there will be 25 '4 times 3,894,- 



B! 4 POWER A2vD PERFOKMAXCE OF EXGEfES. 

730, or 98,926,142 foot-pounds exerted each minute. Dividing 
iis ':.- V.. '.'.'.. ~r z.- :"_r 7.:— ^r ezir:el v 7 this ii:^: is 
equal to 2997*7 aetoal horse-power. 

In the diagrams of the * Victoria and -Albert,' it will be re- 
marked there is a greater disparity in the period of the admission 
of the steam than in the case of the diagrams of the 'Ulster,' 
Lrir.^z ±:n '.':.-. v^ves r_:: ^rizr s: ;::;iri:T> s~:. 

Diagram showing wrong setting of Voices. — In fig. 21 are 
given two diagrams, taken from an engine making 200 strokes 
per minute, applied to work the exhausting apparatus employed 
by the Pneumatic Despatch Company to shoot letters and par- 
cels through a tube. These diagrams show that the valve is 
wrongly set, and that at one end of the cylinder the steam is ad - 

fig. SO. 




::_::._-::: riii: z:~:-z:~z :r z:~zz:^z:: zziz^zzzz ::;:?_:"". 

mitted too soon, and at the other end too late. By following the 
TizL~.--iZ.i ILlzzizzi i: "— '. e s—i. :!:.: :'_r fl_:r':z. r iss.ire is 
closed when about half the stroke has been performed, and that 
the steam is admitted in front of the piston when about one- 
:_-:—.'- :: :he s:r:k~ his still :: :r lerfiniri -vh^rrii rli Ir±- 
hand dia gram shows that a considerable part of the stroke has 
been performed before that end of the cylinder begins to get 
steam. The action in this case would be amended by shifting 
round the eccentric The mean pressure on the piston shown 
by these diagrams is only 10*79 lbs. per square inch. 

Diagram showing the necessity of large Ports for high speeds 
of Piston. — Fig. 22 represents two diagrams taken from the 
same engine with the unequal action at the different ends of the 
cylinder corrected. But the diagrams show that the engine has 



DIAGRAMS FROM FAST ENGINES. 



355 



not enough lead in the valves, and, moreover, that the passages 
are too small for the speed with which the engine works. It 
would he an advantage to increase either the width or the 
amount of travel of the valve of this engine, or both ; as also to 
give more lead, so that the steam would be able to attain and 
maintain its proper pressure at the beginning of the stroke, and 
until it is purposely cut off. The mean pressure of steam on the 
piston shown by the diagrams represented in fig. 22 is 13'36 lbs. 
per square inch. 

Diagrams illustrative of the action of the Link Motion. — 
In fig. 23 we have a diagram taken from a horizontal engine, 
with 27-inch cylinder and 3-feet stroke, constructed by Messrs. 
Boulton and Watt, employed to work the Portsmouth Floating 

Fig. 22. 




DIAGRAMS FROM ENGINE OF PNEUMATIC DESPATCH COMPANY. 



Bridge. The steam is cut off by the link so as to make the ad- 
mission almost the least possible, so as to test the engine itself 
before the chains which draw the bridge backward and forward 
had been applied. With the steam cut off thus early there is 
necessarily a very large amount of expansion, and also a very 
large amount of cushioning ; and it will be observed that the 
steam begins to be compressed at not much less than half-stroke. 
With this amount of expansion the link is 2£ inches from the 
centre. The pressure of steam in the boiler was 22 lbs., and 
that of the vacuum in the condenser 11 lbs. per square inch, 
when this diagram was taken ; and the engines ran without the 
chains at 40 revolutions per minnte. 

Fig. 24 is another diagram taken from the same engine with 



356 



POWER AND PERFORMANCE OF ENGINES. 



the link in the same place. Pressure of steam in boiler, 21 lbs. 
per square inch; pressure of vacuum in condenser, 11 J lbs. per 
square inch; number of revolutions per minute, 35. In this 
diagram, and also in the last, we have a small loop formed at 
the top of the diagram, from causes already explained. 

In fig. 25 we have another diagram taken from the same en- 
gine, but in this case the steam is not shut off by the link but 
by the throttle-valve, and there is consequently very little 
cushioning, and the loop at the top of the diagram almost dis- 

Fig. 23. 




diagram from engine op portsmouth floating bridge. 
(engine throttled by link.) 



appears. "When the diagram was taken the pressure of steam 
in the boiler was 22 lbs., and of the vacuum in the condenser 
Hi lbs. per square inch, and the number of revolutions per 
minute was 38. 

Figs. 2G, 27, and 28 are diagrams taken by Eichards' indi- 
cator from Allen's engine, in the United States department of 
the International Exhibition of 1862. In this engine the diam- 
eter of the cylinder was 8 inches ; length of stroke, 24 inches ; 
pressure of steam in boiler, 49 lbs. per square inch : revolutions 
per minute, 150. 



DIAGRAMS FROM FAST ENGINES. 



357 



Diagrams illustrative of action of Air-pump and Hot-well. 
-Fig. 29 is a diagram taken from the air-pump of the Duke oi 



Fig. 24. 




Ik 



-10 



DIAGRAM FROM ENGINE OF PORTSMOUTH FLOATING BRIDGE. 
(ENGINE THBOTTLED BY LINK.) 

Sutherland's yacht ' Undine,' a vessel fitted with two inverted 
angular engines, with cylinders 24 inches diameter and 15 inches 

Fig. 25. 




TIAGRAM FROM ENGINE OF PORTSMOUTH FLOATING BRIDGE. 
(ENGINE THEOTTLED BY THROTTLE-VALVE.) 



358 



POWER AND PERFORMANCE OF ENGINES. 



stroke. When this diagram was taken the ordinary amount ol 
injection was on, and the engine was working at moorings at 
72 strokes per minute. There was also an air-vessel on the 
hot-well. In fig. 80 we have a diagram taken from the air-pump 
of the same engine, with an extra amount of injection put on. 

Fig. 26. 




DIAGRAM FROM ALLEN'S ENGINE. 

The pump appears to he quite too small for the work it has to 
do, as is seen "by the different configuration of the diagram from 
that of the diagrams represented in figs. 11 and 18, which are 
also diagrams taken from air-pumps. In those diagrams, how- 
ever, the stroke of the "bucket is more than half performed, he- 
Figs. 27 and 28. 




DIAGRAMS PROM ALLEN S ENGINE. 

fore the pressure rises ahove the atmospheric line ; whereas in 
fig. 30, the pressure rises ahove the atmospheric line the moment 
the "bucket hegins to ascend, showing that at that time the 
<*-hoie of the pump harrel is filled with water. The vacuum 
must always he inferior where the air-punip is gorged with 
water, 



DIAGRAMS TAKEN FROM AIR-PUMPS. 



359 



Unlike the previous diagrams taken from air-pumps, we see 
m these figures the pressure or resistance has to he encountered 
from the heginning, or nearly the beginning of the stroke ; and 
the vacuum is not good, and the pump overloaded. There is a 

Fig. 29. 




DIAGEAM FROM AIR-PUMP OP DUKE OP SUTHERLAND S YACHT. 
(OEDINAEY INJECTION.) 

worse vacuum with the increased injection than with the ordi- 
nary injection, showing that it is not the too great heat of the 
condenser which makes the vacuum bad, but a deficient capacity 
of pump, or an imperfect emptying of it every stroke. 



Fig. 30. 



TlSrO 




DIAGRAM FROM AIR-PUMP OF DUKE OF SUTHERLAND'S YACHT. 

(extea injection put on.) 

In fig. 31 we have a diagram illustrative of the diminished 
load upon the air-pump, caused by putting an air-vessel on the 
hot-well, a is the atmospheric line, and b is the line represent- 



360 



POWER AND PERFORMANCE OF ENGINES. 



ing the ordinary pressure existing in the hot-well when the air- 
vessel is in operation. By letting out the air the pressure rises 
to o, showing that the pressure on the pump is less with the air- 
vessel than without it. If the air-vessel be discarded, an in- 
creased velocity must be given to the water passing through the 
waste-water pipe to enable the bucket to ascend, and this im- 
plies a waste of power. 

Fig. 31. 




DIAGRAM FROM HOT-WELL OF DUKE OF SUTHERLAND'S YACHT. 
(AIE- VESSEL ON.) 

In fig. 32 we have a diagram taken from the hot-well of the 
Duke of Sutherland's yacht after the air-vessel has been re- 
moved. In this diagram the pressure begins to rise pretty 
quickly, as the bucket of the pump ascends ; and the maximum 
pressure, when reached, is maintained pretty uniform to the end 
of the stroke. It does not then, however, suddenly fall, but 
only gradually, owing to the momentum of the water ; and the 



Fig. 32. 



Ibs-rs 




DIAGRAM TAKEN FROM HOT-WELL OF DUKE OF SUTHERLAND S YACHT. 
(ATE- VESSEL OFF.) 

pencil does not again come down to the atmospheric line until 
nearly half the downward stroke of the pump has been com- 
pleted. 

In fig. 33 we have a diagram taken from the hot-well of th<? 
steamer ' Scud,' a vessel fitted with two single-trunk engines, 
that is, trunk engines with the trunks projecting only at one 
end, and not at both, as in Messrs. Penn's arrangement. The 



DIAGRAMS TAKEN FROM WATER-PUMPS. 



361 



engines are angular, working up to the screw-shaft, and the 
cylinders are G8 inches diameter, and 4|- feet stroke. The 
trunks are 41 inches diameter. These engines made 42 strokes 
per minute, and worked up to 8-£ times the nominal power. 
The diagram shows an increase of pressure in the hot-well at 

Fig. 33. 



DIAGRAM TAKEN FROM HOT- WELL OF STEAMER ' SCUD.' 

each end of the stroke of the double-acting pump, and the 
pressure runs up slowly at each end of the stroke, when it 
slowly falls, forming the loop shown in the diagram. 

Diagram from Pump of Water-worTcs. — Fig. 34 is a diagram 
taken from the pump of a pumping-engine at the Cork TTater- 

Fig. 34. 




A A 



Q A 



DIAGRAM TAKEN FROM PUMP OF CORK WATER-WORKS. 

works. This engine, in common with most pumping-engines of 
modern construction, is a rotative engine— an innovation first 
effectually introduced by My. David Thomson. The engines 
make 31 revolutions per minute, and work with steam of 40 lbs. 
on the square inch. When the plunger is ascending, the pump 
16 



362 POWER AND PERFORMANCE OF ENGINES 

is sucking ; and when the piston is descending it is forcing, and 
the diagram shows that both operations are accomplished with 
much regularity, and without any of those sudden fluctuations 
which always occasion a loss of power. 

Having now shown in what manner the indicator may he ap- 
plied to ascertain the performance of ordinary engines, I shall 
proceed to describe the manner of its application in the case of 
double-cylinder engines. In this class of engines the steam 
having pressed the first piston to the end of its stroke in the 
manner of a high-pressure engine, escapes, not into the atmos- 
phere, but into another engine of larger dimensions, where it 
expands, and acts as low-pressure steam on the piston of the 
second engine, being finally condensed in the usual manner. 
The pressure urging the first, or high-pressure piston, is conse- 
quently the difference of pressure between the steam in the 
boiler and that in the second cylinder; and the pressure urging 
the second, or low-pressure piston, is the difference of pressure 
between the steam on the eduction side of the high-pressure 
cylinder and that of the vapour in the condenser. There will be 
a small difference between the pressures in the communicating 
parts of the high and low-pressure engines, just as there is a 
small difference between the vacuum in the cylinder and that 
in the condenser. But in well-constructed high-pressure engines 
this difference will not sensibly detract from the power. 

Diagrams from Double-cylinder Engines.— In proceeding to 
determine the power of a double-cylinder engine, we first de- 
termine by a diagram and a computation, such as I have already 
given examples of, the power exerted by the high-pressure en- 
gine ; and then, in like manner, we determine the power exerted 
by the low-pressure engine. The total power is obviously the 
sum of the two. 

An example of the diagrams taken from the high and low- 
pressure cylinders of a double-cylinder engine, at the Lambeth 
Water-works, constructed by Mr. David Thomson, and erected 
under his direction, will next be given. In a paper read by Mr. 
Thomson before the Institution of Mechanical Engineers, and a 
copy of which he has forwarded to me, the main particulars of 



OF THE DOUBLE CYLINDER KIND. 363 

these engines are recited ; and some of the most material points 
of that paper I shall here recapitulate, as these engines consti- 
tute a very superior example of the double- cylinder class of 
engine. 

These engines are beam-engines, having the double cylinders 
at one end of the beam, and a crank and connecting-rod at the 
other end. Tour engines of 150 horse-power each are fixed side 
by side in the same house, arranged in two pairs, each pair 
working on to one shaft, with cranks at right-angles, and a fly- 
wheel between them. The strokes of the crank and of the large 
cylinder are equal ; while the small cylinder, which receives the 
steam direct from the boiler, has a shorter stroke, and its effec- 
tive capacity is nearly one-fourth that of the large cylinder. The 
pumps are connected direct to the beams near the connecting- 
-od end by means of two side rods, between which the crank 
works. The pumps are of the combined plunger and bucket 
construction, and are thus double-acting, although having only 
two valves. This kind of pump, which is now in general use, 
was first introduced by Mr. Thomson at the Eichmond and the 
Bristol Water- works in the year 1848. The following are the 
principal dimensions of the engines : — Diameter of large cylin- 
der, 46 ins. ; diameter of small cylinder, 28 ins. ; stroke of large 
cylinder, 8 ft. ; stroke of small cylinder, 5 ft. 6f ins. ; diameter 
of pump-barrel 23f ins. ; diameter of pump-plunger, 16$ ins.; 
stroke of pump, 6 ft. llf ins. ; length of beam between extreme 
centres, 28 ft. 6 ins. ; height of beam-centre from floor, 21 ft. 
4 ins. The valves are piston-valves, connected by a hollow pipe, 
through which the escaping steam passes, and are so constructed 
that one valve effects the distribution of the steam in each pair 
of cylinders. 

The cylinder-ports are rectangular, with inclined bars across 
the faces to prevent the packing-rings of the valve from catching 
against the edges of the ports ; and the bars are made inclined 
instead of vertical, in order to avoid any tendency to grooving 
the valve-packing. The openings of the port extend two-thirds 
round the circumference of the valve in the ports of the large 
cylinder ; but they extend only half round in the ports of the 



364 POWER AND PERFORMANCE OF ENGINES. 

small cylinder. The packing of the valve consists of the fou? 
cast-iron rings, which are cut at one side exactly as in an ordi- 
nary piston, the joint being covered by a plate inside. A con- 
siderably stronger pressure of the rings against the valve-chest 
is required than was at first expected, because the openings of 
the steam ports extend so far round the valve ; and for this pur- 
pose springs are placed inside the packing-rings to assist their 
own elasticity. This construction of valve has the advantage of 
admitting of great simplicity in the castings of the cylinders ; 
and also allows of the whole of the valve-work being executed 
in the lathe, which is generally the cheapest and most correct 
kind of work in an engineering workshop. These valves are 
worked by cams. 

The principal object aimed at in the construction of this 
piston-valve was a reduction to a minimum of the loss of pres- 
sure which the steam undergoes in passing from the small cyl- 
inder to the large one. This is here accomplished by making 
the passage of moderate dimensions and as direct as possible ; 
and also by preventing any communication of this passage with 
the condenser, so that when the steam from the small cylinder 
enters the passage, the latter is already filled with steam of the 
density that existed in the large cylinder at the termination of 
the previous stroke. In constructing the engines some doubt 
was entertained as to the best size of passage, in order on the 
one hand to avoid throttling the steam, and on the other to ob- 
viate as much as possible the loss of steam in filling the passage. 
The size adopted was a pipe 6 inches in diameter, or l-60th of 
the area of the large cylinder, for a speed of piston of 230 feet 
per minute in the large cylinder : and this is believed to be 
about the best proportion, the entire cubic content of the whole 
passage in the valve amounting to 3,944 cubic inches. The indi- 
cator diagrams show that with this construction of valve there 
is very little or no throttling of the steam, and also that there is 
but a very moderate drop in the pressure as the steam passes 
from the small cylinder into the large one. In this respect the 
valve completely answered the expectations entertained of it 
and left little further to be desired on this point. 



DIAGRAMS FROM DOUBLE CYLINDERS. 



365 



In figs. 35 and 36 we have diagrams taken simultaneously 
from the top of the small cylinder and the bottom of the large 
one, in the double-cylinder engines of the Lambeth Water- 
works, designed by Mr. Thomson — the high-pressure diagram 
being placed above, and the low-pressure diagram below, with a 
small space between the two answering to the loss of pressure in 
the communicating pipe. The dotted line shows the exhaust- 
line in the small cylinder reversed, so as to tell by direct measure 

Fiacs. 35 and 36. 




DIAGRAMS FROM DOUBLE-CYLINDER ENGINES, LAMBETH WATER-WORKS. 
(TAKEN SIMULTANEOUSLY FROM TOP OF SMALL CYLINDER AND BOTTOM OF J.ABGH 

CYLINDER.) 

ment between this bottom and the top of the diagram what is 
the pressure of the steam on the small piston at every part of 
its stroke. 

The most material of the results which may be deduced from 
the indicator diagrams of this engine are as follows :— Percent- 
age of stroke at which steam is cut off in small cylinder, 40 per 
cent. ; total expansion at end of stroke in small cylinder, in 
terms of bulk before expansion, 2*41 per cent. ; amount of ex- 
pansion on passing from small to large cylinder, in terms of bulk 



566 POTTER AND PERFORMANCE OF ENGINES. 

before escaping from small cylinder, 118 per cent. ; total expan- 
sion at end of stroke in large cylinder, in terms of original bulk, 
9 - 66 per cent. ; total amount of efficient expansion, in terms of 
original bulk, 8'19 per cent. ; total pressure of steam per square 
inch at point of cutting off, 41 lbs. ; theoretical total pressure 
at end of stroke of small piston, IT'O lbs. ; actual total pressure 
shown by diagram, 18'0 lbs. ; excess of actual over theoretical 
in percentage of actual pressure, 6 per cent. ; theoretical loss of 
pressure in passage from small to large cylinder, 2*6 lbs. ; actual 
loss shown by diagram, 4*5 lbs. ; theoretical total pressure at 
end of stroke of large piston, 4*2 lbs. ; actual total pressure shown 
by diagram, 5*5 lbs. ; excess of actual over theoretical in per- 
centage of actual pressure, 23 per cent. ; mean pressure on 
crank-pin from both cylinders, 22,400 lbs. ; maximum ditto, 36,- 
058 lbs. ; ratio of maximum to mean, 1*61 to 1*00 ; ratio of max- 
imum to mean pressure on crank-pin in a single cylinder engine 
with the same total amount of efficient expansion, the clearances 
and ports bearing the same proportion to the working capacity 
of the cylinder, namely, l-40th part (this ratio is calculated from 
the ordinary logarithmic expansion curve), 2 - 75 to 1*00; effi- 
ciency of steam contained in large cylinder at end of stroke, as 
shown by diagram, if used without expansion, taken as 1'00; 
actual efficiency of same steam as employed in both cylinders, 
as shown by diagram, 2 - 90; theoretical efficiency of the same 
steam if expanded to the same degree as the total amount of 
efficient expansion, 3*10. The engines are fitted with steam- 
jackets, and these indicator diagrams show that the pressure of 
the steam at the end of the stroke, instead of falling short of 
what it ought to be by the theoretical expansion curve, exceeds 
that amount by about 23 per cent, of the actual final pressure. 
It might be supposed that the increased pressure at the end of 
the stroke was due to the heat imparted from the jackets either 
superheating the steam or converting the watery vapour mixed 
with it into true steam ; and probably the latter is the cause of 
a small part of the observed effect ; but ALr. Thomson considers 
it less likely that sufficient heat could be communicated from 
the jackets to produce an increase of 23 per cent, in the actual 



FEATURES OF WATER-WORKS ENGINES. 367 

final pressure, especially as on several occasions the condensed 
water from the jackets has been collected and found not to ex- 
ceed half-a-gallon per hour. The experiments made on the 
quantities of water passed from the boilers give uniformly the 
result, that a considerably larger quantity of water passes from 
the boilers than is accounted for by the indicator diagrams, 
taking the quantity and pressure of the steam just before it 
escapes to the condenser as the basis of calculation. In some 
trials made within a few days of these diagrams being taken, the 
excess of water thus disappearing from the boilers was about 
37 per cent. To suppose that the valve was leaking might ac- 
count for it;* but besides great care having been taken to avoid 
this source of error, it can hardly be supposed that the valve 
was always leaking more than the pistons. 

To ascertain the amount of friction in these engines Mr. 
Thomson made many experiments, and found that, when the 
engines were new, and working at perhaps little more than half 
their power, the loss in comparing the work done with the indi- 
cator diagrams amounts to as much as 25 per cent, of the indi- 
cated power ; but in these cases the pistons have been too tight 
in the cylinders, and when this error has been corrected, and the 
engines worked up to their regular work, all the losses were 
brought down to from 12 to 15 per cent, of the indicated power. 
This includes the friction of both the engines and the pumps, 
the working of the air-pumps, feed-pumps, cold-water pumps, 
and pumps for charging the ah'- vessels with air. 

"With regard to the economy of fuel attained by these double- 
cylinder engines, it maybe stated that the fourpumping-engines 
at the Lambeth Water-works are fixed in one house, and are 
employed in pumping through a main-pipe 30 inches diameter 
and about nine miles in length; and when all the engines are 
working together at their ordinary speed of 14 revolutions per 
minute, the lift on the pumps, as measured by a mercurial gauge, 
is equal to a head of about 210 feet of water. Under these cir- 
cumstances they were tested by Mr. Field soon after being fin- 

* Some of the disappearance of the heat is no doubt imputable to its truisfor 
(nation into power, as explained under the head of thermo-dynamics. 



368 POWER AND PERFORMANCE OF ENGINES 

ished, in a trial of 24 hours' duration without stopping. Th« 
actual work done by the pumps during this trial was equal to 
97,064,894 lbs., raised one foot high for every 112 lbs. of coal 
consumed ; in addition to which this consumption included the 
friction of the engines and pumps, and the power required to 
work the air-pumps, feed and charging-pumps, and the pumps 
raising the water for condensation. The coal used was Welsh, 
of good average quality. 

The economy in consumption of fuel during this trial, and in 
the subsequent regular working of these engines, together with 
the satisfactory performance generally of the engines and pump 
work, induced the Chelsea Water-works Company, and also the 
New Eiver Company, each to erect in 1854 a set of four similar 
engines, which were made almost exactly the same as the Lam- 
beth Water-works engines already described, with the exception 
that a jacket of high-pressure steam was in these subsequent en- 
gines provided under the bottoms of the cylinders, which had 
not been done with the previous engines. The pumps were 
also different in size to suit the different lifts. 

The New Eiver engines were tested soon after being com- 
pleted, and the result reported was 113 million lbs. raised one 
foot high by 112 lbs. of "Welsh coal. But this duty was obtained 
from a trial of only seven or eight hours' duration, which is too 
short to obtain very trustworthy results. 

The set of engines made for the Chelsea Water-works was the 
last finished, and on completion the engines were tested by Mr. 
Field in the same manner as the Lambeth engines, by a trial of 
24 hours' continuous pumping. The coal used was Welsh, as be- 
fore, and the duty reported was 103*9 million lbs. raised one foot 
high by 112 lbs. of coal. This, as in the previous instance, was 
the duty got from the pumps in actual work done, no allowance 
being made for the friction of the engines and pumps, and the 
power required to work the air-pumps, cold-water pumps, &c. 
At the time of these engines being tested, the loss by friction 
and by working the air-pumps, &c, averaged about 20 per cent. 
of the power, as given by the indicator diagrams ; so that if the 
duty had been estimated from the indicator diagrams, as is usual 



OF THE DOUBLE CYLINDER KIND. 3G9 

in marine engines, it would have been 103*9 x ^f, or about 130 
million lbs. raised one foot by 112 lbs. of coal, which is equiva- 
lent to a consumption of 1*7 lb. per indicated horse-power per 
hour. 

In figs. 37 and 38 we have diagrams taken from a small en- 
gine called "Wenham's double-cylinder engine, working with a 
pressure of 40 lbs. per square inch in the boiler, and exhibited at 
the Great Exhibition in 18C2. The average pressure on the 

Fig. 3T. 





DIAGRAM FROM HIGH-PRESSURE CYLINDER OP WENHAM'S DOUBLE-CYLINDER 

ENGINE. 
(CYLINDER THESE INCHES DIAMETER AND TWELVE INCHES STROKE.) 

piston of the high-pressure engine, which is 3 inches diameter 
and 12 inches stroke, is 26*6 lbs. per square inch, and the pow&r 
it exerts is 3'16 horses. The average pressure exerted on tho 

Fig. 38. 



DIAGRAM FROM LOW-PRESSURE CYLINDER OF WENHAM S DOUBLE-CYLINDER 

ENGINE. 

piston of the low-pressure engine is 8-5 lbs. per square inch, and 
the power it exerts is 2 - 37 horses. The steam in passing from 
one cylinder to the other is heated anew, as had previously been 
done by me in the engines of the steamer 'Jumna,' of 400 horse- 
power. The total power developed in both cylinders of "Wen- 
ham's engine is 6'05 horses. 

Having now explained how to interpret a diagram, the next 
thing is to explain how to take one, and here I cannot do better 
15* 



370 n^rz a>~ zzzziztPavcz <:i zy^iyzs. 

than recite t&e iastractioiis for this operation issued with 

BizliriY in ii :-i:-: r ": 7 :~_t ~-i>e:*5, Zlli:: 1:::1~:-. :: :lr 7-:: ;.- 1. 

Tojfe ike Jfaper. — Tate the outer cylinder off 1 from the instrument, 

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METHOD OP TAKING A DIAGRAM. 371 

rarely happens. The length of the diagram drawn at high speeds should 
not exceed four and a-half inches, to allow changes in the length of the 
cord to take place to some extent, without causing the drum to revolve 
to the limit of its motion in either direction. On the other hand, the 
diagram should never be drawn shorter than is necessary for this 
purpose. 

To take tlie Diagram. — Every thing being in readiness, turn the han- 
dle of the stop-cock to a vertical position, and let the piston of the in- 
dicator play for a few moments, while the instrument becomes warmed. 
Then turn the handle horizontally to the position in which the commu- 
nication is opened between the under side of the piston and the atmos- 
phere, hook on the cord, and draw the atmospheric line. Then turn 
the handle back to its vertical position, and take the diagram. When 
the handle stands vertical, the communication with the cylinder is wide 
open, and care should be observed that it does stand in that position 
whenever a diagram is taken, so that this communication shall not be 
in the least obstructed. 

To apply the pencil to the paper, take the end of the longer brass arm 
with the thumb and forefinger of the left hand, and touch the point as 
gently as possible, holding it during one revolution of the engine, or 
during several revolutions, if desired. There is no spring to press the 
point to the paper, except for oscillating cylinders ; the operator, after 
admitting the steam, waits as long as he pleases before taking the dia- 
gram, and touches the pencil to the paper as lightly as he chooses. Any 
one, by taking a little pains, will become enabled to perform this opera- 
tion with much delicacy. As the hand of the operator cannot follow the 
motions of an oscillating cylinder, it is necessary that the point be held 
to the paper by a light spring, and instruments to be used on engines 
of this class are furnished with one accordingly. 

Diagrams should not be taken from an engine until some time after 
starting, so that the water condensed in warming the cylinder, &c, 
shall have passed away. "Water in the cylinder in excess always distorts 
the diagram, and sometimes into very singular forms. The drip-cocks 
should be shut when diagrams are being taken, unless the boiler is 
priming. If when a new instrument is first applied the line should 
show a little evidence of friction, let the piston continue in action for a 
short time, and this will disappear. 

As soon as the diagram is taken, unhook the cord ; the paper cj-lin- 
der should not be kept in motion unnecessarily, as it only wears out the 
spring, especially at high velocities. Then remove the paper, and 
minute on the back of it at once as many of the following particulars as 
you have the means of ascertaining, viz. : 

The date of taking the diagram, and scale of the indicator. 



372 POWER AXD PEEFOEItfANCE OF ENGINES. 

The engine from which the diagram is taken, which end, and which 
engine, if one of a pair. 

The length of the stroke, the diameter of the cylinder, and the num- 
ber of double strokes per minnte. 

The size of the ports, the kind of valve employed, the lap and lead 
of the valve, and the exhaust lead. 

The amount which the waste-room, in clearance and thoroughfares, 
adds to the length of the cylinder. 

The pressure of steam in the boiler, the diameter and length of the 
pipe, the size and position of the throttle (if any), and the point of cut- 
off. 

On a locomotive, the diameter of the driving-wheels, and the size of 
the blast orifice, the weight of the train, and the gradient, or curve. 

On a condensing-engine, the vacuum by the gauge, the kind of con- 
denser employed, the quantity of water used for one stroke of the en- 
gine, its temperature, and that of the discharge, the size of the air-pump 
and length of its stroke, whether single or double acting, and, if driven 
independently of the engine, the number of its strokes per minute, and 
the height of the barometer. 

The description of boiler used, the temperature of the feed-water, the 
consumption of fuel and of water per hour, and whether the boilers, 
pipes, and engine are protected from loss of heat by radiation, and if 
so, to what extent. 

In addition to these, there are often special circumstances which 
should be noted. 

Counter and Dynamometer. — There are other instruments 
besides the indicator for telling the performance of an engine — 
the counter -which registers the number of strokes made by an 
engine being used for this purpose, in the case of pumping- 
engines, working with a uniform load, and the dynamometer 
being employed in testing the power exerted by small engines. 
The dynamometer consists of a moving disc well oiled, and en- 
circled by a stationary hoop, which can be so far tightened as to 
create sufficient friction to constitute the proper load for the en- 
gine. The hoop is prevented from revolving with the disc by an 
arm extending from it, which is connected with a spring, the 
tension on which, reduced to the diameter of the disc, represents 
the load which the friction creates ; and the load multiplied by 
the space passed through per minute by any point on the cir- 
cumference of the disc will represent the power. Such dyna- 



NEW FORM OF DUTY METER. 



373 



mometers, however, cannot be conveniently applied to large en- 
gines ; and as in steam-vessels, where economy of fuel is most 
important, the counter will not accurately register the work 
done, seeing that the resistance is not uniform, and as without 
some reliable means of determining the power produced in (Lif- 
erent vessels relatively with the fuel consumed, it is impossible 
to establish such a comparison of efficiency as will lead to emula- 
tion, and consequent improvement, I have felt it necessary to 
contrive a species of continuous indicator, or power-meter, for 

Fig. 40. 




BOURNE'S duty meter. 

ascertaining and recording the amount of work done by any 
engine during a given period of time. The outline of one form 
of this instrument is exhibited in fig. 40 ; but I prefer that the 
cylinder should be horizontal instead of vertical, and that it 
should be larger in diameter, and shorter — this figure being 
copied from a photograph of an instrument I had converted from 
a common M'Naught's indicator, for the sake of readiness of 
construction. In this instrument one end of the indicator cylin- 
der communicates with one end of the main cylinder, and the 
other end of the indicator cylinder with the other end of the 



374 POWER AND PERFORMANCE OF ENGINES. 

main cylinder, so that the atmosphere does not press upon the 
piston of the indicator at all, but that piston is pressed on either 
side by steam or vapour of precisely the same tension as that 
which presses on either side of the piston of the engine. The 
indicator piston is pressed alternately upward and downward 
against a spring in the usual manner. A double-ended lever vi- 
brating on a central pivot, and with a slot carried along it near- 
ly from end to end, as in the link of a common link-motion, is 
attached tc the side of the cylinder, and from this slot a horizon- 
tal rod extends to the arm of a ring encircling a ratchet-wheel, 
there being a number of pawls in this ring of different lengths to 
engage the ratchets. This link is moved backwards and forwards 
on its centre, 8 or 10 times every stroke of the engine, by means 
of the lower horizontal rod which is attached at one end to the 
lower end of the link, and at the other end to a small pin in the 
side of a drum, which is drawn out by a string, like the drum 
for carrying the paper in a common indicator, and is, in like 
manner, returned by a spring ; but the dimensions of the drum, 
and the place of attachment of the string, are such that the drum 
makes a considerable number of turns — say 10 — for each stroke 
of the engine, and the link makes the same number of recipro- 
cations. If there be an equality of pressure on each side of the 
piston, the end of the rod moving in the slot will be in the mid 
position ; and as while it is there no amount of vibration of the 
link will give it any end motion, there will be no motion under 
such circumstances communicated to the ratchet. If, however, 
the pressure either upward or downward is considerable, the 
end of the rod will be moved so much up or down in the link 
that its reciprocation will give considerable end motion to the 
rod communicating with the ratchet ; and the amount of motion 
given to the ratchet every stroke will represent the amount of 
mean pressure urging the piston. The number of revolutions to 
be made by the drum every stroke having been once definitively 
fixed, it is clear that the number of revolutions it will make per 
minute will depend on the number of strokes made per minute 
by the engine, and the revolutions of the ratchet-wheel will 
consequently represent both the mean pressure and the speed of 



HEATING SURFACE IN MODERN BOILERS. 375 

piston — or in other words, it will represent the power. The 
spindle of the ratchet wheel is formed into a screw, which works 
into the periphery of a wheel that gives motion to other wheels 
and hands, like the train of a gas-meter ; and on opening the in- 
strument at the end of any given time, such as at the termina- 
tion of a voyage of an ocean steamer, the power which the ves- 
sel has exerted since she started on the voyage will be found to 
be accurately registered. This being compared with the quan- 
tity of coals consumed, which can easily be found from the books 
of the owners, will give the duty of the engine ; and by ascer- 
taining and publishing the duty of different vessels, a wholesome 
emulation would be excited among engine-makers and engine 
tenders, and a vast reduction in the consumption of fuel would 
no doubt be obtained. For many years past I have urged the 
introduction of that system of registration in the case of steam- 
vessels which in the case of the Cornish engines speedily led to 
such unprecedented economy. But the want of a suitable register- 
ing apparatus constituted a serious impediment, and I have con- 
sequently undertaken to contrive the instrument of which a 
rough outline is given above. 

Heating Surface in modern Boilers. — The quantity of heat- 
ing surface given in modern boilers per nominal horse-power has 
been constantly increasing, until, in some of the boilers of recent 
steam-vessels intended to maintain a high rate of speed, it has 
become as much as 35 square feet per nominal horse-power; and 
such vessels exert a power nine times greater than the nominal 
power. The nominal power, in fact, has ceased to be any measure 
of the dimensions of a boiler ; and the best course will be to con- 
sider only the water evaporated. In modern marine boilers it 
may be reckoned that a cubic foot of water will be evaporated 
in the hour by 7 lbs. . of coal burned on 70 square inches of fire- 
bars, and the heat from which is absorbed by 10 square feet of 
heating surface, so that the consumption of coal per hour, on each 
square foot of grate, will be 14*4 lbs. If the steam be cut off 
from the cylinder when one-third of the stroke has been per- 
formed, as is a common practice, the efficiency of the steam 
will be somewhat more than doubled, or a horse-power will be 



576 POWER A2sD PERFORMANCE OF ENGINES. 

generated with something less than S\ lbs of coal. In large 
boilers and engines, however, the efficiency is greater than in 
small, and there is further benefit obtained from superheating, 
and from heating the feed-water very hot. In modern steam- 
vessels of efficient construction, therefore, the consumption of coal 
is not more than 2h lbs. per actual horse-power. Boulton and 
"Watt put sufficient lap upon their valves to cut off the steam 
when two-thirds of the stroke have been performed as a minimum 
of expansion ; and then, by aid of the link-motion, they can ex- 
pand still more, if required, so as to cut off when one-third of 
the stroke has been performed. 

The area of the back uptake should be 1 5 square inches per 
cubic foot evaporated ; the area of the front uptake 12 square 
inches, and the area of the chimney 7 square inches per cubic foot 
evaporated. These proportions will enable the dimensions of any 
boiler to be determined when the rate of expansion has been fixed. 

The proportion in which the actual exceeds the nominal 
power varies very much in different engines, but about 4 or 44 
times appears to be the prevalent proportion in 1865, though, as I 
have stated, in special cases twice this proportion of power is 
exerted, and the boilers are proportioned to give the increased 
supply of steam required. For any temporary purpose the power 
maybe increased by quickening the draught through the furnace 
by a jet of steam in the chimney; but in such case the consump- 
tion of fuel per cubic foot of water evaporated will be somewhat in- 
creased. The first proportion of heating surface, however, which 
the flame encounters is very much more efficient than the last 
portion, in consequence of the higher temperature to which it is 
subjected; and if the draught be quickened the temperature will 
be increased, and every square foot of heating surface will thereby 
acquire a greater absorbing power. The hotter the furnace is, 
the more heat will be absorbed by the water in the region of the 
furnace ; and the more heat that is absorbed by the furnace the 
less will be left for the tubes to absorb. It is material, therefore, 
to maintain high bridges, a rapid draught, and all other aids to a 
high temperature in the furnace ; as the absorption of heat will 
thus be more rapid, and the combustion will be more perfect, 



ADVANTAGES OF HOT FURNACES. 377 

from the high temperature to which the smoke is exposed. It 
will increase the efficacy of the heating surface, moreover, if the 
smoke be made to strike against instead of sliding over it ; and 
this end will be best attained by using vertical tubes, with the 
water within them, on which the smoke may strike on its way 
to the chimney. Such tubes, furthermore, are eligible in con- 
sequence of the facilities they give for the rapid circulation of 
the water within the boiler; and this rapid circulation will not 
merely render the boiler more durable by preventing overheat- 
ing of the metal, but as the rapidly ascending current, by carry- 
ing off the steam and presenting a new surface of water to be acted 
upon, keeps the metal of the tubes cool, they are in a better con- 
dition for absorbing heat from the smoke than if the metal had 
become overheated from the entanglement of steam in contact 
with it, which impeded the access of the water, and prevented 
the rapid absorption of heat which would otherwise take place. 
In locomotive boilers, where the temperature of the furnace is 
very high, as much evaporative efficacy is obtained from 7 lbs. of 
coal, with 5 or 6 square feet of heating surface, as is obtained in 
land and marine boilers with 9 or 10 ; and the reason manifestly 
is, that as the rapidity of the transmission of heat increases as the 
square of the temperature, a square foot of heating surface in a fur- 
nace twice as hot will be four times more effective, so that the 
tubes are left with comparatively little work to do, from so much of 
the work having been done in the furnace. Each square foot of 
tube surface in locomotives will only evaporate as much as each 
square foot in an ordinary land and marine boiler ; but the mean 
efficacy of the whole heating surface is, nevertheless, raised very 
high by the greatly increased efficacy of the fire-box surface, 
from its high temperature. It is desirable to imitate these con- 
ditions in marine and land furnaces by making the area lire-grate 
small, the draught rapid, and the bridges high, to the end that o 
high temperature in the furnace may be preserved, and a con 
sequently rapid generation of steam promoted. It would also be 
desirable, and not difficult, to feed the furnaces with hot air instead 
of with cold, which would conduce more to economy than feeding 
the boiler with hot instead of cold water ; and it would not be dif- 



578 POWER AND PERFORMANCE OP ENGINES. 

ficult to carry out this improvement, by encircling the chimney 
with air- casing nearly to the top, and conducting the air which 
would be admitted by openings around the casings at its upper end, 
past the smoke-box doors, to the end of the furnaces. The only diffi- 
culty which might be apprehended from this procedure would be 
the increased heat and diminished durability of the furnace-bars. 
But this difficulty might no doubt be surmounted by making the 
bars deep and thin, and by not increasing the temperature of the 
entering air beyond the point which experience proved it could be 
raised to with impunity. The area of the casing around the chim- 
ney would require to be about as great, at the largest part, as the 
area of the chimney itself. But it could be made conical, or 
tapering off at the top, and the air might be admitted in vertical 
slits extending downwards for a certain length, as the heat at the 
top of the chimney could be abstracted by such a small volume 
of air as a narrow casing would contain. In this heating of the air 
entering furnaces there is an expedient of economy available for 
the engineer which has not yet been brought into force ; and its 
effect will be both to reduce the consumption of the fuel and to 
render the existing heating surface more effective. If, for ex- 
ample, we take the existing temperature of the furnace to be 
3,000° Fahrenheit, and if we increase the temperature of the en- 
tering air by 500°, which we might easily do without any new 
expense, we shall not merely save one-sixth of the fuel, but wo 
shall render the absorbing surface of the furnace more efficacious 
by raising the temperature from 3,000° to 3,500°. Nor w r ill 
this probably be the limit of benefit obtained ; and as in feeding 
boilers with boiling water instead of cold, and in surrounding 
cylinders by steam to keep them hot instead of exposing them 
to the atmosphere, Ave obtain a greater benefit than theory would 
have led us to expect, so in feeding furnaces with hot air instead 
of cold air we shall in all probability obtain a larger benefit than 
zhat which theory indicates. The experience already obtained 
of the saving effected by using the hot blast in iron smelting 
furnaces certainly points to the probability of such a realization ; 
and one manifest effect will be, that the combustion of the coal 
will be rendered more perfect, and less smoke will be produced. 



DESIDERATA AT THE PRESENT TIME. 379 

Tlie present system of laud and marine boilers, however, is 
altogether faulty, and must be changed completely. When I 
planned and constructed the first marine tubular boiler in 1838, 
and wnich was adapted for working with a high pressure of 
steam, and which also had the advantage of surface condensation, 
the innovation was a step in advance, and it has proved successful 
and serviceable, though up to the present time the system then 
propounded! by me has not been fully wrought out in practice. 
But we now want something much better than what would have 
sufficed for our wants in 1838, and I will here briefly recapitulate 
what we require and must obtain. First, then, we must have a 
still higher pressure of steam than I contemplated in 1838 ; to 
obtain which wren safety we must have two things ; a very strong 
boiler, and absolute immunity from salting. The expedient of 
surface condensation, which I propounded in 1838, as the means 
of accomplishing the last disideratum, though effectual for the 
purpose, and now widely adopted, is less eligible for moderate 
pressures than the method of preventing salting which I have 
since suggested, and which consists in the introduction of a 
small jet in the eduction-pipe, the water of which, though unable 
wholly to condense the steam, will be itself raised to the boiling 
point, and be transmitted to the boiler without any means of 
stopping it off; and the excess of feed- water which, under this 
arrangement, will always be entering the boiler, will escape 
through a continuous blow off, and thus prevent the boiler from 
salting. The column of steam escaping to the condenser will, 
under suitable arrangements, itself force this water into the 
boiler ; and in locomotives, in like manner, the water may bo 
forced into the boiler by using a portion of the steam escaping 
from the blast pipe for that purpose, whereby the boiler will be 
fed with boiling water by the aid of steam otherwise going to 
waste. In this way marine boilers may be kept from salting ; for 
the sulphate of lime which is deposited from sea water at the tem- 
peratures of high-pressure steam, may be separated by filtration 
in the feed pipe. On the whole, for high pressures a small sur- 
face condenser with auxiliary jet seems best. To give a rapid 
circulation to the water, and render the heating surface efficient 



380 POWER AND PERFORMANCE OF ENGINES. 

in the highest degree, the tubes should be upright with the watei 
within them ; and the furnaces should be fed with coal by self-' 
acting mechanism, which would abridge the labor of firing, and 
insure the work being better done. To reduce the strain on the 
engine at the beginning of the stroke, when steam of a high 
pressure is employed, the stroke should be long, the piston small 
in diameter, and a considerable velocity of piston should be em- 
ployed; or, where there are two engines, the steam may be ex- 
panded from the cylinder of one engine into the cylinder of the 
other engine, according to Nicholson's system, whereby twice the 
expansion will be obtained with only the same apparatus. 

Relative surface areas of Boilers and Condensers. — The 
evaporative power of land and marine boilers per square foot of 
heating surface, depends very much upon the structure and con- 
figuration of the boiler. In some marine engines a performance 
of six times the nominal power has been obtained with a propor- 
tion of heating surface in the boiler of only 12 square feet per 
nominal horse-power ; and as about half of this power was ob- 
tained by expanding the steam, 1 cubic foot of water was evap- 
orated by every 4 square feet of heating surface, which is a 
smaller proportion even than that which obtains commonly in 
locomotives. In such cases the proportion of cooling surface in 
the condenser has been made equal to the amount of heating 
surface in the boiler ; and the amount of cooling surface in the 
condenser relatively to the amount of the heating surface of the 
boiler should manifestly have reference to the activity of that 
heating surface. So in like manner it should be influenced by 
the amount of expansion which the steam undergoes in the cyl- 
inder ; since the steam, in communicating power, parts with a 
corresponding quantity of heat. A still more important condi- 
tion of the action of the condenser is, that the water shall pass 
through the tubes with rapidity, and that it shall flow in the op- 
posite direction to the steam, so that the hottest steam shall 
meet the warmest water ; as warm water will suffice to condense 
hot steam, which would be quite inoperative in condensing at- 
tenuated vapour. A common proportion of condenser surface 
in modern engines is "75 that of the boiler surface. Thus a 



INTERNAL CORROSION OF BOILERS. 381 

boiler with 20 square feet of heating surface will have 15 square 
feet of heating surface. But the largest part of this surface is 
required to obtain the last pound or two of exhaustion ; and it 
is preferable to employ a moderate surface to condense the bulk 
of the steam, and to condense the residual vapour by a small jet 
of salt water let in from the sea. It is found advisable to admit 
a small quantity of salt water on other grounds. For the fresh 
water in the boiler, as it forms no scale, leaves the boiler subject 
to the corrosive influence produced by placing a mass of copper 
tubes — on which the sea water acts chemically— in connexion 
with the mass of wet iron which constitutes the boiler ; and, as 
in Sir Humphrey Davy's arrangement for protecting copper 
sheathing by iron blocks, the copper tubes are protected at the 
expense of the boiler, since the communicating pipes and the 
water within them form an efficient connexion. It would be 
easy to break the circuit so far as the metal is concerned by in- 
terposing glass flanges between the flanges of the pipes. But 
this would not stop the communication by the water itself, and 
the best course appears to satisfy the corroding conditions by 
placing blocks of zinc within the condenser, which might be 
corroded instead of the tubes or the boiler. The present anti- 
dote to the corrosive action consists in the introduction of a cer- 
tain proportion of salt water into the boiler, which is intended 
to shield the evaporating surfaces from corrosive action by de- 
positing a coating of scale upon those evaporating surfaces. 
But in this arrangement we have necessarily an excess of water 
entering the boiler ; for we have not only all the water returned 
which passes off as steam, but a certain proportion of sea water 
besides. It will consequently be necessary to provide for the 
excess being blown out of the boiler ; and the question is, whether, 
as we must introduce such an arrangement, it would not be ad- 
visable, with iOW pressures, to make the proportions such as 
would enable us to dispense with the surface condenser alto- 
gether ? If it is retained at all, it should only be retained in 
such shorn proportions as to condense the grossest part of the 
eteam — the water resulting from which should be sent into the 
boiler quite hot, and the rarer part of the steam should be con- 



382 POWER AND PERFORMANCE OF ENGINES. 

densed by a jet of salt water of about the same dimensions as 
that already en ployed. It is very necessary to be careful in the 
case of surface condensers to prevent any leakage of air, which, 
if mingled with the steam, would form a wall of air against the 
refrigeratory surface, which would prevent the contact of the 
gteam and hinder the condensation, precisely as it was found to 
do in the old engines of JSewcomen, where air was purposely 
admitted to form a stratum between the hot steam and the cold 
cylinder ; and which diminished the loss from the condensation 
of the steam within the cylinder to a very material extent. 

Example of modern marine engine and ooiler. — As an exam- 
ple of the proportions of marine engines and boilers and con- 
densers of approved modern construction, I may here recapitu- 
late the main particulars of the machinery of the screw steamer 
'Rhone,' constructed for the West India Mail Company by the 
Mill wall Iron Company in 1865. 

These engines are on the inverted cylinder principle of 500 
horse-power. There are two cylinders of 72 inches diameter 
and 4 feet stroke, and the estimated number of revolutions per 
minute is 52. The cylinders are supported on massive hollow 
standards resting on a bed plate of the same construction. There 
are two air-pumps wrought by links and levers from two pins on 
the ends of the piston rods. The surface condenser is placed 
between the two air-pumps, and is fitted with brass tubes placed 
horizontally, and resting in vertical tube plates. The two end 
plates have screwed stuffing boxes, with cotton washer packing 
for each tube. The tubes are divided into three groups or sec- 
tions, through each of which the condensing water successively 
passes ; and the water enters from the lower end of the con- 
denser and escapes at the upper end, where the steam enters, so 
that the hottest water meets the hottest steam. The two circu- 
iting pumps are placed opposite each other, and are wrought by 
a crank on the end of the crank shaft. The steam is condensed 
outside the tubes ; and the condensed water flows down to the 
air-pumps, by which it is puinped to the hot well, and from 
which it is taken to the boilers in the usual way. 

The crank shaft is of Krupp's cast steel in two pieces, 



PRINCIPLE OF GIFFARD's INJECTOR. 383 

coupled by flanges. The screw shaft is of iron, covered with 
brass in the stern tube, and working in lignum vitas bearings in 
tho stern tube and after stern post. The boilers are in four 
separate parts, and fitted with a superheating apparatus consist- 
ing of a series of vertical iron tubes 4^ inches bore, on the plan 
of Mi*. Eitchie, the company's superintending engineer. 

The surface condenser has 3,566 tubes, f inches external 
diameter, and 9 feet 2i inches long between the tube plates. 
The surface of the tubes is 6,525 square feet, or 13*05 square feet 
per nominal horse-power. The two circulating pumps are dou- 
ble acting 25" diameter, with a trunk of 17" diameter on one 
end of the plungers. The boilers have 20 furnaces 3' 0|" wide, 
with fire bars oi 6 feet 8 inches in length. The total area of 
fire grate is 400 square feet, = 0*8 square feet per nominal horse- 
power. The number of brass tubes in the boiler is 1,180 of 3^ 
external diameter and 6 feet 8 inches long. The total heating 
surface in the boilers is 9,800 square feet, or 19'6 square feet per 
nominal horse-power. In the superheater the surface is 2,160 
square feet or 4*32 square feet per nominal horse-power, making 
the total heating surface in boiler and superheater 23 '92 square 
feet per nominal horse-power. The area of heating surface in 
the boiler per square foot of grate is 24'5 square feet, and the 
area of superheating surface per square foot of grate is 5*4 square 
feet, making the total heating surface in boiler and superheater 
29'9 square feet per square foot of grate. The total area of the 
condenser surface is *68 of the total heating surface in the boiler, 
and *54 of the total area of the heating surface of boiler and 
superheater taken together. These engines are very strong, and 
manifestly embody the results of the long experience of steam 
navigation which the "West India Mail Company must now pos- 
sess. The workmanship and materials are of the very first 
quality ; and accurate adjustment and conscientious construc- 
tion are manifested throughout. 

Giffard's Injector. — This instrument, which feeds boilers by 
a je* of steam discharged into the feed pipe, acts on the principle 
that the particles of water which obtain a high velocity when 
they flow out as steam retain this velocity when reduced by 



384 POTTER AND PERFORMANCE OF ENGLNES. 

condensation to tlie form of water ; and a jet of water of great 
velocity is capable of balancing a correspondingly high bead, or 
a pressure greater tban that which subsists within the boiler. 
The jet consequently penetrates the boiler, as we can easily un- 
derstand any jet would do which has a greater velocity than a 
similar jet escaping from the boiler. These injectors, though 
very generally employed in locomotives, are not much used for 
land or marine boilers ; and in their present form they occasion 
much waste, as the steam by which they are actuated is drawn 
from the boiler, whereas it ought to be the steam, or a portion 
of it, which escapes to the condenser or the atmosphere. These 
injectors, like Bom-don's gauges, and other instruments employed 
in the steam-engine, are not made by engineers, but are a dis- 
tinct manufacture; and the manufacturers, on being supplied 
with the necessary particulars, furnish the proper size of instru- 
ment in each particular case. The proper diameter of the nar- 
rowest part of the instrument to deliver into the boiler any given 
number of gallons per horn*, may be found by dividing the num- 
ber of gallons required to be delivered per hour by the square 
root of the pressure of the steam in atmospheres, and extracting 
the square root of the quotient, which, multiplied by the con- 
stant number '0158, gives the diameter in inches at the smallest 
part. Contrariwise, if we have the size, and wish to find the 
delivery, we multiply the constant number 63'4 by the diameter 
in inches and square the product, which, multiplied by the square 
root of the pressure of the steam in atmospheres, gives the de- 
livery in gallons per hour. These rules correspond very closely 
with the tables of the deliveries of different sizes published by the 
manufacturers, Messrs. Sharp, Stewart, and Co., of Manchester. 

POWEE EEQTIEED TO PEEFOEil VAEIOrS KTXDS OF WOEFJ. 

The power required to obtain any given speed in a given 
steamer will be so fully discussed in the next chapter that the 
subject need not be further referred to here ; and in my ' Cate- 
chism of the Steam-Engine ' I have recapitulated the amount of 
power, or the size of engine, required to thrash and grind corn, 
spin cotton, work sugar and saw mills, press cotton, drive piles. 



EFFICIENCY OF HYDRAULIC MACHINES. 385 

dredge earth, and blow furnaces. The subject, however, is so 
important that I shall here recapitulate other cases for the most 
part derived from experiments made with the dynamometer in 
France by General Morin,* whose researches on this subject 
have been highly interesting, and have been conducted with 
much care and ability. 

Comparative efficiency of different machines for raising water. 
—Of the different pumps experimented upon by General Morin, 
the result of eight experiments made with pumps draining mines 
showed that the effect utilised was 66 per cent, of the power 
expended. But in these cases there was considerable loss from 
leakage from the pipes. At the salt works of Dreuze the useful 
effect was 52*3 per cent, of the power expended. In fire-engine 
pumps employed to deliver the water pumped at a height of 
from 12 to 20 feet, the proportion of the water delivered to the 
capacity of the pump was, in the pumps of the following makers 
— Merryweather, Tylor, Perry, Oarl-Metz, Letestu, Flaud, and 
Perrin, respectively, as follows :— -920, '887, -910, '974, -910, -920, 
and *900 ; while the percentage of useful effect relatively with 
the power expended was 39*7, 39*1, 30*2, 28*7, 27*1, 19*4, and 
15 *5, respectively. "With a higher pressure, the efficiency of the 
whole of the pumps increased ; and when employed in throwing 
water with a spout-pipe the delivery of water relatively with 
the effective capacity, or space described by the piston, was, when 
the names are arranged, as follows : — Carl-Metz, Merryweather, 
Tylor, Letestu, Perry, Maud, Perrin, and Lamoine, respective- 
ly, -950, -810, -565, -870, '910, -912, -950, and -900 ; while the 
proportion of useful effect, or percentage of work done relatively 
with the power expended, was 80, 57*3, 54*5, 45*2, 37*8, 33-4, 
28*8, and 17*5, in the respective cases. In the membrane pump 
of M. Brule the efficacy was found to be 40 to 45 per cent, of the 
power expended. In the water-works pumps of Ivry, construct- 
ed by Cave, the efficiency was found to be 53 per cent, of the 
power expended ; and in the water- works of St. Ouen, by the 
same maker, 76 per cent. It is desirable that the buckets of 
the pumps of water-works should move slowly, otherwise the 

* Aide Mi.rn.oire \>y General Morin, 5ta edition, 1864. 

17 



386 POWER AND PERFORMANCE OP ENGINES. 

water will go off with considerable velocity', involving a corre- 
sponding loss of power. The area through the valves should be 
half the area of the pump, and the area of the suction and forcing 
pipes ought to be equal to three-fourths of the area of the body 
of the pump. "VFaste spaces should be avoided. The loss of water 
through the valves before they shut is, in good pumps, about 10 
per cent. 

In a chain-pump the efficiency was found to be 38 per cent., 
but in many chain-pumps the efficiency is much more than this. 
The efficiency of the Persian wheel was found to increase very 
much with the height to which the water was raised. For 
heights of 1 yard it was 48 per cent., for 2 yards 57, for 3 yards 63, 
for 4 yards 60, and for 6 yards and upwards 70 per cent, of the 
power consumed. For a wheel of pots the efficiency is 60 per 
cent. ; Archimedes screw, 65 per cent. ; scoop wheel with flat 
boards moving in a circular channel, 70 per cent. ; improved 
bucket wheel, 82 per cent., and tyrnpan-wheel, or, as it is some- 
times called, "Wirtz's Zurich machine, 88 per cent. This machine 
should dip at least a foot into the water to give the best results. 
In the belt-pump the efficiency was found to be 43 per cent.; in 
Appold's centrifugal pump, 65 per cent. ; in the centrifugal 
pump, with inclined vanes, 42 per cent., and with radial vanes, 
24 per cent. In Gwynn's pump the efficiency was 30 per cent. 

In the Archimedes screw the diameter is usually one-twelfth 
of the length, and the diameter of the newel or central drum 
should be one-third of the diameter of the screw. It ought to 
have at least three convolutions, and the line traced by the 
screw on the enveloping cylinder should have an angle of 67° to 
70° with the axis. The axis itself should make an angle of from 
30° to 45° with the horizon. There is a sensible advantage ob- 
tained from working hand-pumps by a crank instead of a lever. 

Old French Flour Mill at Senelle. — Diameter of millstones, 
70 inches ; number of revolutions per minute, 70 ; quantity of 
corn ground and sifted per hour, 260*7 lbs. ; power consumed, 
3 '34 horses. The power is in all cases the power actually exert- 
ed, as ascertained by the dynamometer. 

English Flour Mill near Metz. — Diameter of millstones, 51 18 



POWER REQUIRED TO DRIVE MILLS. 387 

inches; number of revolutions per minute, 110; weight of mill- 
stones, 1 ton ; corn ground per hour by each pair, 220 lbs. ; with 
two pairs of millstones acting, one bolting machine and one win- 
nowing machine, the power consumed was 8|- horse-power. 

English Flour Mill near Verdun. — Diameter of millstones, 
51*18 inches; number of revolutions per minute, 110; quantity 
of corn ground per hour by each pair, or by each revolving mill- 
stone, 220 lbs. ; with two stones revolving the power consumed 
was 5*64 horses. The power consumed by one winnowing ma- 
chine and two bolting machines, with brushes sifting 1,650 lbs. 
of flour per horn*, was 6 1 horses. In another mill the number of 
turns of the millstone was 486 per minute, the quantity of corn 
ground by each horse-power was 120 lbs., and the quantity of 
corn ground per hour was 110 lbs. of which 72*7 per cent, was 
flour, 7*3 per cent, was meal, and 19*5 per cent, was bran. In a 
portable flour-mill, with machinery for cleaning and sifting, the 
total weight was 1,000 lbs. 

Barley Mill. — Number of revolutions of the millstone per 
minute, 246 ; barley ground per hour, 143*68 lbs. ; motive force in 
horses, 3*11 ; barley ground per hour by each horse-power, 48*2 
lbs. The products were, of first and second quality of barley 
flour, 60*12 per cent., of meal and bran, 30*25 per cent., and of 
bran and waste, 9*63 per cent. 

Bye Mill. — Number of revolutions of the millstone per minute, 
448 ; rye ground per hour, 92*114 lbs. ; power expended, 2*86 
horses; temperature of flour, 60*8° Fahr. ; products 64*9 per 
cent, of flour, 9*1 per cent, of meal, and 26 per cent, of bran. In 
another rye mill the revolutions of the millstones per minute 
were 232 ; rye ground per hour, 180 lbs. by 2*19 horse-power, 
and the rye ground per hour by each horse-power was 82*21 
lbs. The products were 72*5 per cent, of flour; 17*5 per cent, 
of meal and fine bran, and 10 per cent, of bran and waste. 

Maize Mill. — Number of revolutions of the millstone per 
minute, 246 ; maize ground per hour, 73*96 lbs. ; motive force in 
horses, 2*69 ; maize ground per hour by each horse-power, 27*5 
lbs. Products: first and second quality of flour, 61*1 per cent.; 
meal and fine bran, 30*2 per cent. ; bran and waste, 4*7 per cent. 



388 POWER AND PERFORMANCE OP ENGINES. 

Vermicelli Manufactory. — External diameter of edge runners^ 
66'93 inches; internal diameter of edge runners, 62*99 inches; 
number of revolutions of the arbour of the mill per minute, 4 ; 
pounds of paste prepared per hour, 77 lbs. ; power expended, 2*95 
horse-power. 

Bean Mill. — dumber of revolutions of the millstone per min- 
ute, 496; power expended per hour, 1*76 horse. 

Oil Mill. — "Weight of edge runners, 6,600 lbs. ; number of 
turns of the vertical spindle per minute, 6 ; weight of seed intro- 
duced every ten minutes, 55 lbs. ; weight of seed crushed daily, 
3,300 lbs. ; product in oil in 12 hours, 1,320 2bs. ; power expend- 
ed, 2*72 horses. 

Saw Mill — Weight of the saw frame, 842*6 lbs. "When cut- 
ting dry oak 8.73 inches thick, with 1 blade in operation, the 
reciprocations or strokes of the saw were, 88 per minute, the 
surface cut, '525 square foot, and the power expended 3*3 horses. 
"When cutting the same wood with 4 blades in operation, the 
number of strokes of the saw per minute was 79 ; the surface cut by 
each per minute *483 square foot, or 1*73 square foot per minute 
for the 4; and the power expended was 3*70 horses, which is 
equivalent to 28 square feet cut per hour by 1 horse-power. "When 
cutting four-year seasoned oak, 12*4 inches thick, with 4 blades, 
making 90 strokes per minute, the surface cut by each blade was 
"35 square foot, and the surface cut by the 4 blades, 1*41 square 
foot. "When the saw was run along the middle of a cylindrical 
log of beech one-year cut, 23*6 inches diameter, the number of 
strokes of the saw per minute was 88; the surface cut per minute, 
•968 square foot ; and the power expended, 3 horses. In these 
experiments the breadth of the saw cut was "157 inch, and the 
experiments show that it does not take more power to drive a 
frame with one saw than to drive a frame with four, the great- 
est part of the power indeed being consumed in giving motion to 
the frame. The common estimate in modern saw mills, when 
the frame is filled with saws, is, that to cut 45 superficial feet of 
pine, or 34 of oak per hour, requires 1 indicated horse-power. 
Ilie crank, which moves the frame up and down, and which is 
nsually placed in a pit under the machine, should have balance 



POWER REQUIRED TO DRIVE SAWS. 389 

weights applied to it, the momentum of which weights, when 
the saw is in action, will be equal to that of the reciprocating 
frame. In some cases the weight of the saw frame is borne by 
a vacuum cylinder, and with a 20-inch stroke it makes 120 
' strokes per minute. 

Circular Saw. — Diameter of saw, 27*5 inches ; thickness of 
oak cut, 8*73 inches; number of revolutions per minnte, 266; 
surface cut per minute, 1*93 square foot; power consumed 3*55 
horses. When set to cut planks of dry fir, 10*62 inches broad, 
and one inch thick, the number of revolutions made by the saw 
per minute was 244; surface cut per minute, 7*67 square feet; 
and the power expended, 7'35 horses. These results show that 
in sawing the smaller class of timber one circular saw will do at 
least as much work as four reciprocating saws, with the same 
expenditure of power. The surface cut is, in all these cases, under- 
stood to be the height multiplied by the length, and not the sum 
of the two faces separated by the saw. The speed of the circular 
saw here given is not half as great as that now commonly em- 
ployed. Circular saws now work with a velocity at the peri- 
phery of 6,000 to 7,000 feet per minute, and band saws with a 
velocity of 2,500 feet per minute, and it is generally reckoned 
that 75 superficial feet of pine, or 58 of oak, will be sawn per 
hour by a circular saw for each indicated horse-power expended. 
Planing machine cutters move with a velocity at the cutting edge 
of 4,000 to 6,000 feet per minute, and the planed surface travels 
forward J fl th of an inch for each cut. 

Reciprocating Veneer Saw. — Length of stroke of saw, 47*24 
inches; thickness of the blade, *01299 inch; breadth of saw cut, 
•02562 inch; length of teeth for mahogany and other valuable 
woods, *196 inch; pitch of the teeth, *3939 inch; distance ad- 
vanced by the wood each stroke, *0196 to '03937 inch; number 
of strokes of the saw per minute, 180 ; surface cut per hour 
counting both faces, 107*64 square feet; power expended 0'66 
horses. 

Sawing Machine for Stones. — Soft sandstone : breadth of saw- 
cut, J inch ; time employed to saw 10 square feet, 5 minutes 25 
seconds ; power expended 4*54 horses. Hard sandstone : breadth 



390 POWER AND PERFORMANCE OP ENGINES. 

of saw-cut, £ inch ; time employed to cut 10 square feet, 1 hour 
87 minutes ; power expended 2 horses. 

Sugar Mill for Canes. — A three-cylinder mill, with rollers 
5J feet long, 30 inches diameter, and making 2£ turns a minute, 
driven by an engine of 25 to 30 horse-power, will express the 
juice out of 130 tons of canes in 12 to 15 hours. An acre of 
land produces from 10 to 20 tons of canes, according to the age 
and locality of the canes. The juice stands at 8° to 12° of the 
saccharometer, according to the locality. The product in sugar 
varies from 6 to 10 per cent, of the weight of the canes, accord- 
ing to the locality and mode of manufacture. "Well-constructed 
mills give in juice from 60 to TO per cent, of the weight of the 
canes, and one main condition of efficiency is, that the rollers 
shall travel slowly, as with too great a speed the juice has not 
time to separate itself from the woody refuse of the cane, and 
mucli of it is reabsorbed. To defecate 330 gallons of juice 6 
boiling-pans, or caldrons, are required, 4 scum presses, and 10 
filters ; and to granulate the sugar 2 vacuum pans, 6|- feet diam- 
eter, are required, with 2 condensers, and it is better also to 
have 2 air-pumps. The steam for boiling the liquor in the 
vacuum pans is generated in three cylindrical boilers, each 6 feet 
in diameter. To whiten the sugar there are 10 centrifugal 
machines, driven by a 12-horse engine, which also drives a pair 
of crushing-rollers. The sugar in the centrifugal machines is 
wetted with syrup, which is driven off at the circumference of 
the revolving cylinders of wire gauze, carrying with it most of 
the colouring matter of the sugar, which to a great extent adheres 
to the outside of the crystals, instead of being incorporated in them, 
and may consequently be washed off. "When the sugar is thus 
cleansed it is again dissolved, and the syrup is passed through 
deep filters of animal charcoal. Provision must be made to 
wash the charcoal, both by steam and by water, and two fur- 
naces, to re-burn the animal charcoal, will be required. 

The action of animal charcoal in bleaching sugar is not well 
understood. But it appears to be due to certain metallic bases 
ui the bones, which by burning are brought to or towards the 
metallic state, from the superior affinity of the carbon present 



EXAMPLE OF A COTTON MILL 391 

for tlie oxygen in the base at the high temperature at which the 
re-burning takes place. "When, however, the charcoal is mixed 
with the syrup, the metallic base endeavours to recover the 
oxygen it has lost, by decomposing the water, leaving thereby a 
certain quantity of hydrogen in the nascent state ; and this hy- 
drogen appears to dissolve the small particles of carbon in the 
sugar which detract from its whiteness, and to form therewith 
a colourless compound. When the metallic basis has recovered 
all its lost oxygen the charcoal ceases to act, and has to be re- 
burned ; and, after numerous re-burnings, the charcoal appears 
to be all burned out of the bones, when re-burning ceases to be 
of service. But their efficacy might be restored by mingling por- 
tions of wood charcoal. The use of charcoal in sugar refining is 
not merely a source of expense in itself, but it occasions a loss 
of sugar, as, when the mass of charcoal becomes effete, it is left 
saturated with syrup, and the water with which it is washed has 
to be boiled down, to recover the sugar as far as possible. I 
consequently proposed several years ago a method of revivifying 
the charcoal without removing it from the filter. But the 
method has not yet been practically adopted. 

The begass, or woody refuse of the cane, is usually employed 
to generate the steam in the boilers. But it is generally neces- 
sary to use coal besides. 

Fans for olowing Air. — The indicated power required to 
work a fan may be ascertained by multiplying the square of the 
velocity of the tips in feet per second by the collective areas of 
the escape orifices in square inches, and by the pressure of the 
blast in pounds per square inch, and finally dividing the product 
by the constant number 62,500, which gives the indicated power 
required. The pressure in pounds per square inch may be de- 
termined by dividing the square of the velocity of the tips in feet 
per second by the constant number 97,300. 

Cotton-spinning Mill. — Number of spindles, 26,000 ; power 
consumed, 110 horses; Nos. of yarn spun, 30 to 40; spindles 
witfc preparation driven by each horse, 237. It is reckoned that 
each machine requires 1 horse-power. 

Another example of a Cotton Mill. — Number of spindles, 



392 POWER AND PERFORMANCE OP ENGINES. 

14,508 ; power required to drive tliem, 50*5 horses; Nos. of yarn 
spun, 30 to 40 ; spindles and preparation driven by eacli horse* 
power, 287. 

Another example of a Cotton Mill. — Number of spindles, 
10,476 ; Nos. of yarn spun, 30 to 40 ; spindles and preparation 
driven by each horse-power, 235. 

Details of power required oy each Machine in Cotton Mills. — ■ 
One beater making 1,100 revolutions per minute, with ventilat- 
ing fan making half this number of revolutions, cleaning 132 lbs. 
of cotton per hour, requires 2*916 horse-power. One beater 
making 1,200 revolutions per minute, with combing drum 1*23 
feet diameter and 2*8 feet long, making 800 revolutions per min- 
ute, and preparing 132 lbs. of cotton per hour, requires 800 revo- 
lutions per minute and 1*767 horse-power. Power required to 
work the fluted cylinders and endless web of this machine, *312 
horse. Twelve double-casing cylinders, with eccentrics, re- 
quiring 2*697 horses, including the transmission of the motion, 
or per machine, *225 horse. Transmitting the motion for 26 
carding-machines requires 1*82 horse-power. One simple card, 
consisting of a drum 39 - 37 inches diameter and 19*68 inches 
long, making 130 revolutions per minute, and carding 2 lbs. of 
cotton per hour, requires *066 horse-power, without reckoning 
the power consumed in communicating the motion. The same 
card working empty requires *044 horse-power. One double- 
carding machine carding 4*18 lbs. of cotton per hour, requires 
*207 horse-power. A drawing-frame drawing 119 lbs. per hour 
requires 1*835 horse-power. A roving-frame, with 60 spindles, 
with cards, making 525 revolutions per minute, and producing 
42 lbs. of No. 7 rovings per hour, requires *760 horse-power. 
One frame with screw-gearing, having 60 spindles, making 550 
revolutions per minute, and producing 42 lbs. of No. 7 per hour, 
requires *486 horse-power. Two frames with screw-gearing, 
each containing 96 spindles, making in one case 510 revolutions 
ind the other 500 revolutions per minute, producing 28*6 lbs. 
of No. 2*75 to 3 per hour, requires 1*482 horse-power. Two 
frames with screw-gearing, one containing 78 spindles making 
344 revolutions per minute, and the other 60 spindles making 260 



TOWER REQUIRED TO DRIVE WOOLLEN MILLS. 393 

revolutions per minute, and producing 57'2 lbs. of No. 8 per 
hour, requiring *797 horse-power. One spinning-frame, with 
cards, having 240 spindles, making 5,000 revolutions per minute, 
and producing 1*65 lb. of yarn of No. 38 to No. 40 per hour, 
requires '686 horse-power, and in another experiment *648 horse- 
power. Three spinning-frames for weft, having each 360 spin- 
dles, making 4,840 revolutions per minute, and producing 8 lbs. 
of No. 30 to No. 40 yarn per hour, require 2*103 horse-power. 
One retwisting machine, with 120 spindles, making 3,000 revo- 
lutions per minute, requires 1*19 horse-power. One dressing 
machine for calico 35|- inches wide, with ventilator : speed of 
the principal arbor, 176 revolutions per minute; speed of the 
brushes, 45 strokes per minute ; power required, *735 horse. 
The same machine, with the ventilator not going, requires *206 
horse-power. 

Power-loom Weaving. — To drive one power-loom weaving 
calico 35J inches wide, and 82 to 90 picks per inch, making 105 
strokes per minute, requires, taking an average of four experi- 
ments, "1195 horse-power. 

Another example of Power-loom Weaving. — Number of 
looms weaving calico driven by water-wheel, 260 ; dressing ma- 
chines, 15 ; winding machines, 5 ; warping machines, 8 ; small 
pumps, 6 ; yards of calico produced per month, 283,392 ; power 
required to drive the mill, 25 - 6 horses; number of looms, with 
accessories, moved by 1 horse, 12. 

Another example of Power-loom Weaving. — Total number of 
looms, 60 ; dressing machines, 5 ; warping machines, 3 ; wind- 
ing machines, 2 ; monthly production of cotton cloth called 
'Normandy linen,' 47^ inches wide, 360 pieces, each 396 yards 
long ; power consumed, 8 horses ; looms with their accessories 
moved by each horse-power, 7'8. 

Wool-spinning Mill. — Machines driven : simple cards, 29 ; 
double cards, 2 ; scribbling beater, 1 ; mules of 240 spindles, 8 ; 
mules of 200 spindles, 4: lathes, 3; power consumed, 9*75 
horses. Also in another experiment with 9 simple and 3 double- 
carding machines, 2 beaters, and 2 scribbling machines, the 
power consumed in driving was 3*5 horses. 
17* 



394 POWER AND PERFORMANCE OF ENGINES. 

Another example of a Wool-spinning Mill. — A wheel exert- 
ing 10 horse-power drives 6 mules of 240 spindles, 6 of 180, 2 
of 192, 2 of 120, and 5 of 100, making in all 3,644 spindles ; also 
32 carding and 2 scribbling machines. Another wheel, also ex- 
erting 10 horse-power, drives 8 mules of 240 spindles, 4 of 120, 
and 7 of 180, making in all 3,660 spindles; also 31 carding and 
2 scribbling machines, and 2 beaters. The spindles, number- 
ing in all 7,304, make 5,000 revolutions per minute, and the 
cards 88 to 89, requiring a horse-power for 365 spindles. Prod- 
uct per day of 12 hours, 1,100 lbs. of yarn from No. 12 to 
No. 13. 

Details of Power consumed in spinning Wool. — One winding 
machine with 16 bobbins, without counting the power expended 
in the transmission of the motion, requires to drive it *259 
horse ; 3 winding machines with 64 bobbins in all, with power 
lost by transmission, 1*427 horse; one mule spinning No. 6 warp 
yarn, with 220 spindles, making 3,650 revolutions per minute, 
•259 horse. One mule called 'Box-organ,' spinning In o. 50 warp 
yarn with 300 spindles, making 3,200 revolutions per minute, 
requires 1*273 horse-power. 

Mill for spinning Wool and weaving Merinos. — Nineteen 
machines to prepare the combed wool, having together 350 
rollers; 16 mules with 3,400 spindles ; one winding machine of 
60 rollers to prepare the warp ; 2 warping machines ; 2 self- 
acting feeders; 100 power-looms; 2 lathes for wood and iron, 
and 1 pump, require in all 30 horse-power. Produce : 13,600 
cops of woollen thread, of 45 cops to the lb., each measuring 
792 yards. The looms make 115 revolutions per minute, and 
produce daily 4 pieces of double- width merino of 68 yards each, 
and 4 pieces of simple merino of 1*2 to 1*4 yard broad, and each 
88 yards long. 

Fulling Mill. — In fulling the cloths called 'Beauchamps,' 
each piece being 220 yards long, and *66 yard wide, and weigh- 
ing from 121 to 127 lbs., the fuller making 100 to 120 strokes 
per minute, each piece requires 2 hours to full it, and the expen- 
diture of 2 horse-power during that time. 

Flax Manufacture. — A machine for retting the flax, having 



POWER REQUIRED TO DRIVE FLAX MILLS. 395 

15 pairs of rollers with triangular grooves, requires 3*376 horse- 
power, and the heckles *057 horse-power. 

One fly breaking-card 12*59 inches diameter and 47*24 inches 
long, making 915 revolutions per minute, with a drum of 42*12 
inches diameter, and 47*24 inches long, making 76 revolutions 
per minute; 4 distributing rollers, having a diameter of 4 inches 
and a length of 47*24 inches, making 380 revolutions per minute ; 
8 travellers, 5 inches diameter and 47*24 inches long, making 10 
turns per minute, and one combing cylinder 15 inches diameter 
and 47*24 inches long, making 6 revolutions per minute, require 
together 1*939 horse-power, and produce 17 lbs. of carded flax 
per hour. 

One finishing carding cylinder, 40 inches diameter and 47'24 
inches long, making 176 revolutions per minute ; 5 distributing 
rollers, 4 inches diameter, making 23 revolutions per minute ; 
4 travellers, 5 inches diameter, making 7'3 revolutions per min- 
ute ; 1 combing cylinder, 15 inches diameter, making 3*4 revolu- 
tions per minute, together require *811 horse-power, and produce 
8|- lbs. of carded flax per hour. 

One spinning-machine, containing 132 spindles, making 2,700 
revolutions per minute, spinning yarns from No. 7 to No. 9, re- 
quires 1*24 horse-power, and produces 3f lbs. of yarn per hour. 

One spinning-machine, having 168 spindles, making 2,700 
revolutions per minute, and producing 3 lbs. of No. 18 to 24 
yarn per hour, requires 1*96 horse-power. 

"Wet spinning of flax : one drawing-frame drawing a sliver 
for No. 20 yarn, requires *493 horse; drawing-frame drawing 
sliver for No. 50 yarn, requires *487 horse ; drawing-frame draw- 
ing sliver for No. 70 yarn, requires '495 horse. 

Second drawing-frame, drawing two slivers for yarns Nos. 
20 and 30, requires *68 horse ; second drawing-frame, drawing 
two slivers for yarns Nos. 30 and 40, requires *544 horse ; second 
drawing-frame, drawing one sliver for No. 60 yarn and one for 
No. 70, requires *617 horse. 

Third drawing-frame, drawing two slivers for yarns Nos. 30 
tc 60, requires *69 horse. 

Eoving-frame of 8 spindles, preparing the flax for yarn No. 



396 POWER AND PERFORMANCE OF ENGINES. 

20, requires *608 horse ; roving-frame of 8 spindles, preparing 
the flax for No. 30 yarn, requires *486 horse; frame of 16 spin- 
dles, preparing the flax for ISTo. 40 yarn, requires *987 horse- 
power. 

Paper Manufacture. — In some cases the pulp, or stuff of 
which paper is made, is obtained by heating the rags by stamp- 
ers ; but more generally it is produced by placing the rags be- 
tween revolving cylinders stuck full of knives. "When produced 
by stampers, the proportions of the apparatus are as follows : 
weight of stampers, 220 lbs. ; distance of the centre of gravity 
from the axis of rotation, 4 feet ; rise of the centre of gravity 
each stroke, 3 J inches ; number of stampers, 16 ; number of 
lifts of each stamper per minute, 55 ; weight of rags pounded in 
12 hours by each stamper, 33 lbs. ; weight of stuff produced in 
12 hours by each stamper, 122 lbs. ; power consumed, 2 - 7 horses. 

Chopping-cylinders, for preparing the pulp : number of cyl- 
inders working, 2 ; number of turns of cylinders per minute, 
220 ; weight of rags chopped and purified in 12 hours, 528 lbs. ; 
power consumed, 4*48 horses. 

In another instance, 10 cylinders for preparing the pulp, 
making 200 revolutions per minute, 1 paper-making machine, 
cutting-machines, pump, and accessories, consumed 50-horse 
power. The machine made 13 yards of paper per minute, and 
the produce was 1 ton of printing paper per day of 24 hours. 

In another instance, 28 pulping-cylinders, and 3 paper-mak- 
ing machines produced 2 to 3 tons of paper per day of 24 hours, 
and consumed 113 horse-power. 

Printing Machinery. — Printing large numbers is now per- 
formed by cylindrical stereotype plates, revolving continuously ; 
and the ' Time3 ' and other newspapers of large circulation are 
thus printed. The impressions are taken from the types in 
papier mache, and in twenty minutes a large stereotype plate 
is ready to be worked from. The power required to drive 
this machine varies with the number of impressions required in 
the hour. For 5,000 impressions per hour, the power required 
is 3*75 horses ; for 6,000 impressions, 4*77 horses; 7,000 impres- 
sions, 5'9 horses; 8,000 impressions, 7'03 horses; 9,000 impress 



WEAVING BY COMPRESSED AIR. 397 

sions, 8*75 horses ; and 10,000 impressions, 10*35 horses. The 
paper should he supplied to such machines in a continuous web, 
with a cutter to cut off the sheets at the proper intervals, and a 
steam cylinder to dry and press them. But this has not yet 
been done. The machine could also be easily made to perforate 
the paper along the edges of the leaves, and to fold each paper 
up and put a print-ed and stamped paper envelope around it, so 
as to be ready at once to put into the post-office or to distribute 
by hand. The most expeditious mode of stereotyping would 
be to use steel types set on a cylinder, against which another 
cylinder of type-metal is pressed, and the paper would then be 
printed in the same manner as calico. 

Glass Works. — Mill to grind red lead : to grind 3 tons, the 
vertical arbor requires to make for the first ton 20 revolutions 
per minute, for the second 25, and for the third 40, consuming 
5'28 horse-power. Vertical millstones, to grind clay and broken 
crucibles; diameter of the granite stones or runners, 3*7 feet; 
thickness, 1*4 foot; weight, 1 ton; distance of edge runners 
from central spindle, 4 feet ; number of turns of the arbor per 
minute, 7|- ; power consumed 1*92 horse. In the 12 hours 6 or 
8 charges of about 300 lbs. each of old glass pots are ground, 
and about 3 tons of dry clay. Wheels for cutting the glass, 170 ; 
*athes for preparing the cutting wheels, 5 ; lathes for. metal, 2 ; 
power consumed, 17'9 horses; wheels driven by each horse- 
power, 9 '5. 

Iron- Works. — The weekly yield of each smelting furnace in 
Wales is from 100 to 120 tons ; pressure of blast, 2|- to 3 lbs. pei 
square inch ; temperature of the blast, 600° Fahr. ; yield weekly 
of each refining-furnace, 80 to 100 tons; of each puddling-furnace, 
18 tons ; of each balling-furnace for bars, 30 tons; of each ball- 
ing-furnace for rails, 80 tons; iron rolled weekly by puddle rolls, 
300 tons ; by rail rolls, 600 tons ; power required to work each 
train of rail rolls, 250 horses ; to work puddle rolls and squeezer, 
80 horses; small bar train, CO horses; pumping air into each 
blast-furnace, 60 horses; into each refining-furnace, 26 horses; 
Tail saw, 12 horses. 

Weaving Tyy compressed air. — In common power-looms, the 



598 POWER AND PERFORMANCE OF ENGINES. 

shuttle is driven backward and forward by a lever which imi- 
tates the action of the arm in the hand-loom. But it has long 
been obvious to myself and others that it might be shot back- 
ward and forward like a ball out of a gun, by means of com- 
pressed air. This innovation has now been practically carried 
out. Bat the benefits derivable from the practice have been 
much exaggerated, and a much more comprehensive improve- 
ment than this is now required. Indeed, reciprocating looms of 
all kinds are faulty, as they make much noise, consume much 
power, do little work, and cannot be driven very fast ; and the 
proper remedy lies in the adoption of a circular loom in which 
the cloth will be woven in a pipe, and in which many threads 
of weft will be fed in at the same time. 

Circular Loom. — The obvions difficulty in a circular loom, is 
to drive the shuttle round continuously within the walls formed 
by the warp. One mode of driving proposed by me, is by mag- 
nets or other suitable form of electro-motive machine, which 
does not require contact ; and the shuttle should be a circular 
ring, with many cops placed in it, so that many threads might 
be woven in at once. The desideratum, however, is to weave a 
vertical pipe with the bobbins of the weft in the centre of the 
circle ; and this may be done by depositing the thread between 
metallic points, like circular heckles, which points will change 
then- positions inward or outward at each time a thread is de- 
posited. These points would conduct the threads of the warp. 



CHAPTER VIL 

STEAM NAVIGATION. 

Stea^i navigation embraces two main topics of enquiry : — 
the first, what the configuration of a vessel shall be to pass 
through the water at any desired speed with the least resist- 
ance ; and the second, what shall be the construction of ma- 
chinery that shall generate and utilise the propelling power 
with the greatest efficiency. The second topic has, in most of 
its details, been already discussed in the preceding pages ; and it 
will now be proper to offer some remarks on the remaining 
portion of the subject. 

The resistance of vessels passing through the water is made 
up of two parts : — the one, which is called the bow and stern 
resistance, being caused partly by the hydrostatic pressure forc- 
ing back the vessel, arising from the difference of level between 
the bow and stern, and partly by the power consumed in blunt 
bows in giving a direct impulse to the water ; while the other 
part of the resistance, and the most important part, is that due 
to the friction of the water on the sides and bottom of the ship. 
The bow and stern resistance may be reduced to any desired 
extent by making the ends sharper. But the friction of the bot- 
tom cannot be got rid of, or be materially reduced, by any means 
yet discovered. 

"When a vessel is propelled through water, the water at the 
bow has to be moved aside to enable the vessel to pass ; and the 
velocity with which the water is moved sideways will depend 
upon the angle of the bow and the speed of the vessel. When 



400 STEAM NAVIGATION. 

these elements are known it is easy to tell with what velocity 
the water will be moved aside; and when we know the velocity 
with which the water is moved, we can easily tell the power 
consumed in moving it, which power will, in fact, he the weight 
of the water moved per minute multiplied by the height from 
which a body must fall by gravity to acquire the same velocity. 
But as nearly all the power thus consumed in moving aside the 
water at the bow of a vessel is afterwards recovered at the stern 
by the closing in of the water upon the run, it is needless to go 
into this investigation further than to determine what amount 
of power is wasted by the operation, or in other words, what 
amount of power is expended that is not afterwards recovered. 

If the vessel to be propelled is of a proper form, each particle 
of water will be moved sideways by the bow, in the same man- 
ner as the ball of a pendulum is moved sideways by gravity, so 
as to enable the vessel to pass ; and when the broadest part of 
the vessel has passed through the channel thus created, each 
particle of water will swing backward again until it comes to 
rest at the stern. There will be no waste of power in this 
operation, except that incident to the friction of the moving 
water ; just as in the swinging of a pendulum there is no expen- 
diture of power beyond that which is necessary to overcome 
the friction of the air upon the moving ball. But as the move- 
ment of the vessel, however well she may be formed, will some- 
ichat raise the water at the bow, and somewhat depress the 
w'ater at the stern, there will be a certain hydrostatic pressure 
required to be continually overcome as the vessel advances in 
her course, which opposition constitutes the bow and stern re- 
sistance ; and this, with the friction of the bottom, make up the 
whole resistance of the ship. Before, however, proceeding to 
investigate the amount of this hydrostatic resistance, it will be 
proper to show how accidental sources of loss may be elim- 
inated from the problem by the introduction of that particular 
form of vessel which will make this resistance a minimum ; and 
T will therefore first proceed to indicate in what way such form 
of vessel may be obtained. 

If we take a short log of wood, such as is shown by the 



FORM OF MINIMUM RESISTANCE. 



401 



dotted lines a b c d e f g, in the annexed figure (fig. 41), and if 
we proceed to enquire in what way we shall mould this log into 
a model which shall offer the least possible hydrostatic resist- 
ance in being drawn through the water, we have the following 
considerations to guide us in arriving at the desired knowledge : 
We shall, for the sake of simplification, suppose that the cross 
section of the completed model is to be rectangular, or in other 
words, that the model is to have vertical sides and a flat bottom ; 
for although this is not the best form of cross section, as I shall 

Fig. 41 




-x 



afterwards show, the supposition of its adoption in this case will 
simplify the required explanations. 

"We first draw a centre line x y longitudinally along the top 
of the model from end to end, and continue the line vertically 
downward at the ends as at y z, which vertical lines will form 
the stem and stern post of the model. At right angles to the 
first; line, and at the middle of the length of the model, we draw 
the line a, which answers to the midship frame ; and midway 
between a and the ends we draw other two lines 5 ~b. We may 
afterwards draw any convenient number of equi-distant cross- 



402 STEAM NAVIGATION. 

lines, or ordinates, as they are termed, that we find to be convert 
ient, Now as, by the conditions of the problem, the particles 
of water have to swing sideways like a pendulum, in order that 
the resistance may be a minimum, the particle which encounters 
the stem at x must be moved sideways very slowly at first, like a 
heavy body moved by gravity, but gradually accelerating until 
it arrives at Z>, midway between x and a, where its velocity will 
be greatest ; and this point answers to the position of the ball 
of the pendulum when it has reached the bottom of the arc, and 
has consequently attained its greatest velocity. Thereafter the 
motion, which before was continually accelerated, must be now 
continually retarded, as it is in any pendulum that is ascending 
the arc in which it beats, or in any ball which is projected up • 
wards into the air against the force of gravity. When the par 
tide of water has attained the position on the side of the model 
which is opposite to the midship frame a, it will have come to 
rest, this being the point answering to the position of the pen- 
dulum at the top of its arc, and when just about to make the 
return beat. Thereafter the particle which was before moved 
outwards, will now move inward with a velocity, slow at first, 
but continually accelerating, until it attains the position on the 
side of the model which is opposite to the frame d, when the 
velocity again begins to diminish ; and the particle finally comes 
to rest at the stern. A particle of water that is moved in this 
way will be moved with the minimum of resistance; for since 
it retains none of the motion in it that has been imparted, but 
surrenders the whole gradually without impact or percussion, 
by the time it has come finally to rest, there can be no power 
consumed in moving it except that due to friction only. "Wher- 
ever the water is not moved in this manner it will either retain 
some of the motion, which implies a corresponding waste of 
power, or heat will be generated by impact, which also involves 
a corresponding waste of power. That the water may be moved 
in the same manner as a pendulum is moved, is obviously possi- 
ble, by giving the proper configuration to the sides of the 
model ; and in fact, if an endless sheet of paper be made to 
travel vertically behind a pendulum, with a pencil or paint 



CURVE OF GRAVITY. 



403 



brush stuck in the "ball, the proper form for the side of the 
model will be marked upon the paper. The curve, however, 
which is a parabolic one, may be described geometrically as 
follows : — 

If we compute the height through which a heavy body fall3 
by gravity in any given number of seconds, we shall find that in 
the first quarter of a second it will have fallen through 1 T ^ foot, 
in the second quarter of a second 3 f \ feet, in the third 9£, in the 
fourth 16^2, in the fifth 25 T 2 o 5 o, in the sixth 36 T 3 ^-, in the seventh 
49 T 4 jj 9 2) in the eighth 64|, in the ninth 81§ J, and in the tenth 
quarter of a second lOOf |. The height fallen through, there- 
fore, or the space described by a falling body in a given time, 

Fig. 42. 



10 



•r 



r ; f ] r t r f r — \~^ 

I "1 I I: : I !nHH^l 

r mt ' ' '- '- - — « 



15 



y 



25 3G 49 Ci SI 100 



varies as the square of the time of falling ; and any body which 
is to be moved in the same manner as a falling body is moved by 
gravity, must have the motion imparted to it gradually at the 
same rate of progression. If, then, we draw a line, x y in fig. 
42, and which line we may suppose to be the vertical plane of the 
keel, then if we form the parallelogram abod, with the line x y 
passing through the middle of it, and make this parallelogram one- 
fourth of the length of the vessel and half the breadth, and divide 
the line x y into any number of convenient parts or ordinates, 
say 10, by the vertical co-ordinates numbered from 1 to 10, then 
if we cause the lengths of these successive and equidistant co- 
ordinates, measuring from the line x y, to follow the same law 
of increase that answers to the height through which a body 



404 STEAM xatigahox. 

falls by gravity in. successive and equal portions of time, a line 
traced through the ends of these different lines will give the 
right form for the side of a vessel to hare, in order that it may 
move the water sideways, in the same manner, or according tc 
the same law, by which a -heavy body falls vertically by gravity ; 
and consequently such line is the proper water-line of a ship form- 
ed under the conditions supposed, in order that it may have a 
minimum resistance. The heights -of the several vertical ordi- 
nates — which are drawn on a different scale from the lengths, 
marked on the line x y, are — 1, 4, 9, 16, 25, 38, 49, 64, 81. 
100, which, it will be seen, are the squares of the horizontal 
ordinates 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 ; and the scale by 
which these vertical ordinates are measured is formed by divid- 
ing the distaace y d, which represents one-fourth of the breadth 
of the vessel, into 100 equal parts. The ordinate y t> is therefore 
equal to 100 of those parts, the next ordinate to 81 of them, the 
next to 64 of them, and so on, until the height vanishes at x 
altogether. We might have divided the line -x y into nine equal 
parts, or into S, or 7, or any other convenient number. In such 
case the vertical line y d would have to be divided into 81 equal 
parts to obtain the vertical scale, or into 64, or into 49, according 
as 9, 8, or 7 had been the number selected ; but the number of parts 
into which y d is divided must always be equal to the square of 
the number of ordinates, or the square of the number of parts 
into which the horizontal line is divided. As it is difficult to 
measure the hundredth part of such a small length as y d, we 
may call the number of parts 10 instead of 100, in which case 
the length of the next ordinate will be 8*1, of the next 6*4, of the 
next 4*9, and so on — the whole of the squares being divided by 10 ; 
which proceeding will in no way affect the result, as, in point of 
fact, the difference is only much the same thing as if we meas- 
ured in inches instead of in feet. 

In the figure, x y is five times longer than y d, and x y rep - 
resents one-fourth of the length of the vessel, and y n one-fourth 
of the breadth. The curved line x d represents the proper form 
of the water-line of the front half of the fore body in the case of 
a vessel of these proportions, and with a rectangular cross-sec- 



CURVE OF GRAVIVY. 



405 



tion. The water-line of the second half of the fore body is 
formed by repeating the same curve, but inverted and reversed, 
this will be made obvious by an inspection of fig. 43, where the 
first half of the fore body is repeated on a smaller scale ; and the 
second portion of the fore body is added thereto, thus continuing 
the water-line to the midship frame a a. Here the rectangle 
enclosing the water-line of the first half of the vessel is shown in 
dotted lines, as is also the rectangle enclosing the water-line of 
the first half of the fore body ; and it is plain that the shaded 
space a d is the exact duplicate of the shaded space x d; so that 
if the figure x d has been obtained, we may obtain the figure d a 
by cutting out of the paper the figure x d, inverting it and re- 
Fig. 43. 




versing it, so that the line x d shall coincide with the line d a, and 
the point x with the point a ; or the figure d a may be con- 
structed by co-ordinates in exactly the same manner as the figure 
x d. If the vertical sides of the vessel be formed with the curve 
shown by the curve line x a, then it will follow that a particle 
of water encountering the stem at #, will be moved aside slowly 
at first, and with a rate continually increasing, like a body falling 
by gravity, until the frame ~b lying midway between the stem and 
the midship frame is reached, at which point the water will be 
moving sideways with its greatest velocity. Thereafter the vessel 
will not move the water, but merely follow up the motion already 
given to it, and as the water, when no longer impelled sideways 
by the vessel, will move slower and slower, and gradually come to 
rest, so the vessel will have less and less following up to do, until 
at the midship frame a a, the side motion of the water ceases 



406 STEAM NAVIGATION. 

altogether. Thereafter the water begins to move in the opposite 
direction to fill up the vacuity at the stern left by the progress 
of the vessel. The water gravitates into the run slowly at first, 
and the velocity increases until the point midway between the 
midship frame and the stern is attained, at which point the ve- 
locity is greatest ; and from thence the velocity of the water, 
flowing inward, continually diminishes, until it comes to rest at 
the stern. 

A rectangular box, such as that shown by the dotted' lines 
abodefg, fig. 41, into which the model exactly fits, is called 
its circumscribing parallelepiped ; and it will be at once appar- 
ent, on a reference to fig. 41, that the bulk or capacity of the 
model is exactly one-half of its circumscribing parallelepiped. 
The rectangle x d is equal to the rectangle d y } and the shaded 
space x d being equal to the shaded space d a, the area included 
between the water-line and the vertical plane of the keel, namely, 
the area x y a, is clearly equal to the rectangle d dy a. But that 
rectangle, and the rectangle standing beneath it, are equal to the 
whole area within the water-line of the fore body, and two 
similar rectangles are equal to the area within the water-line of 
the after body. As these four rectangles form just half the area 
of the circumscribing parallelogram, the total area within the 
water-line is equal to half the area of the circumscribing parallel- 
ogram. But the area multiplied by the depth gives the capacity, 
and as the depth of the model is the same as that of the box, or 
circumscribing parallelepiped, while the area of the circumscrib- 
ing parallelogram is twice that of the area of the figure within 
the water line, it follows that the volume or bulk of the model is 
just one-half of the circumscribing parallelepiped. This forms a 
measure of sharpness which in no case it is useful to exceed, if 
the section be made rectangular, or, in other words, if the vessel 
be built with a flat bottom and vertical sides. But if the vessel 
dq built with a rising floor the effect is equivalent to a reduction 
of the breadth, and the circumscribing parallelepiped would, in 
such case, be that answering to the equivalent breadth. "What- 
ever be the form of the cross-section, however, the sectional area 
at each successive frame should be equal to that of a vessel "with 



FISUES CONFORM TO THE LAW. 407 

a rectangular section having water-lines formed on the principle 
which has been here explained. There are other curves, uo 
doubt, which equally with that described by a pendulum fulfil the 
indication of beginning and terminating the motion gradually so 
as to involve no loss of power, and any of these curves are eligi- 
ble as the water-line of a ship. But the pendulum curve is the 
most readily understood, and the most conveniently applicable 
to practical uses, while it perfectly fulfis the required indica- 
tions. If in any intended vessel we have a given form of cross- 
section, and a given ratio of length to breadth, we can easily 
determine the proper water-lines of such a vessel by taking the 
case of a hypothetical vessel of rectangular cross-section having 

Fig. U. 




the same area of midship-section, and by forming the water-lines 
for this hypothetical vessel on the principle already explained. 
The area of cross-section at each successive frame of this hypo- 
thetical vessel, will be the proper area at each successive frame 
of the intended vessel. It is obvious that, according to the prin- 
ciple here unfolded, the form of water-line must vary with 
every alteration of the cross -section ; and in some cases, although 
the same rate of displacement as that already indicated is pre- 
served, the water-lines will cease to be hollow at any part. Thus 
the cylindrical solid, with pointed ends, shown in fig. 44, is virtu- 
ally of the same form as that represented in fig. 41, since the 
area of each successive circular cross-section is the same as 
those of each rectangular cross-section in fig. 41. This solid is 



408 STEAM NAVIGATION. 

supposed to be wholly immersed. It has, in some cases, been 
made an objection to the use of hollow water-lines for ships, that 
in the case of fishes, however fast swimming, no hollow lines are 
to be found in them. Tig. 44, however, which resembles the 
form of a fish, shows that fishes form no exception to the appli- 
cation of the law of progressive parabolic displacement already 
explained ; and if a fast-swimming fish be cut across at equal 
distances, and the areas of these sections be computed and laid 
down with a rectangular outline of uniform depth, it will be 
found that the skin or covering placed over the ends of these 
sections or frames will assume the very form which has been de- 
lineated in the foregoing figures as that proper for a solid in- 
tended to pass through the water with the least amount of hydro- 
static resistance. 

In fig. 44, x y is the axis of the pointed cylindrical solid; and 
a is the circle or section which answers to the midship frame, 
and T) ~b the sections answering to the frames lying midway be- 
tween the centre frame and the ends. The other lines corre- 
sponding to those marked on the model shown in fig. 41, and the 
area of each successive circle is equal to the area of each successive 
rectangular section of the model delineated in fig. 41. The 
water consequently will be displaced at the same rate by one 
solid as by the other. For actual vessels, with rounded bilges 
and more or less rise of floor, the form of the water-lines will 
be neither that shown in fig. 41 nor fig. 44, but will be some- 
thing intermediate between the two; but such, nevertheless, 
that the transverse sectional area of that part of the vessel 
beneath the water-line shall at each successive frame vary in the 
ratio pointed out. 

As water is practically incompressible by any force which a 
ship can bring to bear upon it, the water which a ship displaces 
must find some outlet to escape ; and it will escape in the line 
of least resistance, which is to the surface. A particle of water, 
therefore, on which a ship impinges, will have two kinds of mo- 
tion — one a motion outwards and inwards, such as has been 
already described as resembling the motion of a pendulum, and 
the other a motion upwards and downwards, caused by the ne- 



REST FORM OF CROSS-SECTION. 



409 



cessity of the particles beneath the surface rising up towards the 
surface to allow the vessel to pass, and afterwards of sinking 
down at the stern to fill the vacuity which the progress of the 
vessel would otherwise occasion. This last motion also resem- 
bles that of a pendulum, the particles of water at the stem rising 
up until they attain their greatest height at the midship frame, 
%nJ then again subsiding towards the stern. 

It is not difficult, from these considerations, to deduce the 
conclusion tnat the form of vessel with a flat floor is not the best 
which can be adopted, as will be more clearly understood by a 
reference to fig. 45, where the rectangle d e f g, represents the 

Fig. 45. 




cross-section beneath the water-line of a flat-floored vessel at the 
point midway between the stem and the midship frame, while 
the triangle a bo is the cross-section of a sharp-floored vessel 
at the same point, and with the same sectiona. area. The 
draught of water in each case is 10 feet, represented by the 
figures 1 to 10 ; and the half breadth of the vessel with the 
rectangular cross-section at this point of the length is 5 feet, which 
also is one-fourth of the midship breadth. As the water has to 
be set back from the line of the stem to the line of the side, or 
in the case of the flat-floored vessel, through a distance of o feet, 
we may represent the power consumed in the operation by 5 
feet multiplied by the mean hydrostatic pressure of the water on 
each square foot. The mechanical power required to be ex- 
18 



410 



STEA1C NAVIGATION. 



landed therefore in separating the water in the two sections will 
be as follows : — 



Eectansular section. 



5 


X 


1 


= 


o 


5 


X 


2 


= 


10 


I 


X 


3 


= 


15 


5 


X 


4 


= 


20 


5 


X 


•:• 


= 


25 


5 


X 


G 


= 


SO 


5 


X 


1 


= 


35 


5 


X 


8 


= 


40 


5 


X 


9 


= 


45 


5 


X 


10 


= 


50 



Triangular section. 
9x1=9 


8 


X 


2 


= 


16 


1 


X 


3 


= 


21 


6 


X 


4 


= 


24 


5 


X 


5 


= 


25 


4 


X 


6 


= 


24 


3 


X 


*l 


= 


21 


2 


X 


8 


= 


16 


1 


X 


9 


= 


9 





X 


10 


— 






275 165 

The area of the triangle abc being equal to that of the 
rectangle deig, the weight of water displaced by a foot in the 
lengthof the vessel will be the same whichever form of cross- 
section is adopted ; and as the areas of the shaded triangles ado; 
and b d x, or of the corresponding triangles bfz and cgx, are also 
the same, they represent equal amounts of outward motion of 
the water, and also equal amounts of displacement. In the one 
case, however, this motion is produced against a much greater 
hydrostatic pressure than in the other case ; and as by shifting 
the triangle b f x into the position o & x— whereby we enable the 
vessel to move outward the same volume of water, but against a 
less hydrostatic resistance— we transform the rectangle hbfg 
into the triangle h b c, it follows that there is less resistance 
caused by the movement of the water in the case of triangular 
cross sections than in the case of rectangular. The rubbing sur- 
face too is less in the triangular section. By the principles of 
geometry, applicable to all right-angled triangles,* (bf) : + (rx) z = 

* This is proved by the 47th Proposition of the first book 
of Euclid, which shows that the area of the square described 
on the side a c, opposite" to the right angle of a right-angled 
triangle is equal to the sum cf the squares described en ths 
other sides a b and b c. 




BEST FORil OF CROSS-SECTION. 



411 



(b x}\ As b f = 5 feet and r x also == 5 feet, then (b f) 2 = 25, 
and (f x) 2 = 25, and 25 + 25 = 50, consequently b x = V 50 = 7 
nearly. The length of the immersed triangular outline is conse- 
quently 7 x 4 = 28 feet, whereas the length of the rectangular 
outline = 3 x 10 = 30 feet. As the resistance due to the friction 
of the bottom varies as the quantity of rubbing surface, it follows 
that, as regards friction, the triangular outline is also the more 
Instead, however, of a simple triangle, it is preferable 



eligible. 



Fig. 46. 




that the cross-section should be of the order of figure indicated 
as the best for the horizontal water-lines; and the same con- 
siderations which led to the conclusion that this form would 
offer the least resistance in the case of a body moving through 
stationary water lead also to the conclusion that it will offer the 
least resistance to water moving upwards past a stationary ob- 
ject — which a ship may be supposed to be relatively to the plane 
in which she floats. Such a figure is represented in fig. 46, in 
which the triangular section is shown in dotted lines, and the 
waving lines pass alternately without and within the dotted lines. 
The cross-section of the vessel is for the most part of the outline 



412 



SIEASI NAVIGATION. 



a semi-circle m m m — a semicircle "being the form which presents 
the smallest perimeter relatively with the immersed sectional 
area; but the triangular portion m n is added both to prevent 
the vessel from rolling inconveniently, and to bring the outline 
into the waving curve which other considerations point out as 
the most eligible. One of these considerations, as already men- 
tioned, is that it best fulfils the condition of beginning the up- 
ward displacement slowly, and another is that it effects the least 
possible alteration in the shape of the displaced water. In 

Kg. 47. 




altering the form of a liquid, as in altering the form of a solid, 
there is a certain expenditure of force ; and although this ex- 
penditure in the case of a liquid is relatively very small, it is 
large enough to be worthy of attention in a case where large 
amounts are consumed in giving motion to water. It hence be- 
comes better, since the displaced fluids must assume the form of 
a wave, to effect the displacement so that this form shall be at 
once acquired, instead of some other form being first given to it 
which is subsequently changed by the action of other forces. 
This reasoning will be better understood by a reference to fig. 



BEST FORM OF CROSS-SECTION. 413 

47, where wl is the ■water-level, m n the cross-section cf half the 
vessel, and a a the wave which would he raised if there were 
no outward motion of the water, but only an upward motion. 
The outward motion reduces the altitude to some such small 
elevation as a a. Nevertheless it is advisable that the outline of 
the wave a a should be the same order of figure as the outline 
of the wave a a, only laterally extended. Such indeed is the 
shape it will necessarily assume ; and there will be less change 
of shape and therefore less motion of the internal particles, if the 
wave a a is drawn out sideways from a block of water of the 
form a a, than if drawn out from a rectangular, triangular, or any 
other form of block. The dotted lines indicate the directions in 
which the pressure will be transmitted, and if we suppose these 
lines to be tubes, it will be obvious that the surface of the water 
in these tubes will only conform to the outline of a wave, if the 
side of the vessel has that outline. If we suppose the portions 
of those tubes rising above the water-line to be very mueh en- 
larged, then the height of the outline will fall from AAto«a, 
but the same order of figure will still be preserved, as it involves 
less expenditure of power to give this form at once than to give 
some other form which is afterwards reduced by,the action of 
gravity to this one, so on this ground it is preferable to make 
the cross-section of the vessel of the form suggested. Taking all 
things into account, a curve of the same kind that has been 
shown to be the best for the water-lines, appears to be also the 
best for the cross-section ; and the same ordinates which answer 
for the water-lines will answer for the cross-section, only in the 
latter case the ordinates must be placed closer together. If, for 
example, we have a vessel 200 feet long, and if the ordinates 
of the water-lines be 5 feet apart, there will be 40 ordinates ; 
and if the vessel be supposed to draw 20 feet of water, the same 
ordinates placed 6 inches apart will give the proper form of the 
cross-section below the load water-line. The nearer the form 
of the cross-section approaches to a semicircle the less friction 
there will be in the vessel ; and the proportions of the cross- 
aection should in all cases, where practicable, approach to the 
proportions of a semicircle, or in other words the depth below 



41 4 



STEAM NAVIGATION. 



the water should be a little more than half the breadth at the 
■water-line. 

The ascending water will move more and more rapidly as it 
comes nearer to the surface, like the motion of a falling body in- 
verted ; and its momentum will carry it above the surface to a 
height equal to that which would generate the velocity. This 
motion of the water above the surface constitutes the second 
half of the beat of the pendulum which each ascending particle 
may be supposed to be — the motion of the particle from the keel 
to the water-line being the first half of such beat. -But as, after 
passing the surface of the water, the particle has to encounter 
more of the power of gravity, whereas below the water line it is 
floated by the other contiguous particles, it will follow that the 

Fig. 49. 




motion of the particle above the surface will be smaller in the 
proportion of the greater retarding force it there has to encoun- 
ter. This action will be better understood by a reference to 
fig. 48, where the parallelogram a b o d is supposed to be the 
side of a ship, w l is the surface of the water in which the ship 
swims, and the vertical dotted line at a shows the position of the 
midship frame. If we suppose a particle of water to be situated 
at a? a little below the water-level at the bow, then as the vessel 
moves onward in the direction of the arrow, such particle will 
be moved upwards faster and faster, until midway between the 
bow and the midship frame, where its velocity upwards is great- 
est, it will rise above the surface of the water w l, and its own 
momentum and that of other ascending particles will carry it 
upwards until it reaches the position of the midship frame, when 
it wUl begin to sink, until at y it reaches the same level from 



MEASURE OF THE HYDROSTATIC RESISTANCE. 415 

which it rose. The surface particles, no doubt, which terminate 
their motion at ?/, begin it at w and not at x, and to this circum- 
stance we may trace the origin of the hydrostatic resistance of 
the bow. The depression at y will be as great below the mean 
water-level w l as the elevation at a is above it ; and if the sur- 
face of the water at the stem stood at x instead of at w, the fore- 
body would be in equilibrium, seeing that the depression txw 
would suck the vessel forward as much, or nearly so, as the pro- 
tuberance from t to a would impede it. As the hydrostatic 
pressure from a to s pushes the vessel forward as much as the 
depression from s to y holds it back, the two portions of the after 
body will be in equilibrium ; and the whole moving vessel would 
be in equilibrium if the surface of the water at the stem stood at 
x instead of at w. As, however, the water stands higher at the 
stem than at the stern, there will be a hydrostatic resistance to 
be encountered which is equal to the height of the wave midway 
between a and w, which will be \a, acting against the breadth 
of the ship. This will readily be understood by a reference to 
fig. 49J-, which represents a horizontal slice of a floating body of 
the height of the wave which the body raises in passing through 
the water, and the form of the wave is represented by the trian- 
gular figure w a c, which is delineated on the plane surface 
formed by cutting away one-quarter of the model so as to clear 
the problem of the complication involved by the introduction of 
the curved form of the side. A transverse ordinate is drawn at 
5, and at the point 5 &, where this ordinate meets the side, a line 
is drawn parallel to the axis, intersecting the line c e. From the 
point of intersection a vertical line b is raised, on which is set 
off the height of the wave at 1> &, and by drawing any desired 
number of similar lines the wave w a c will be set off on the 
midship section in the form e e d, which figure represents the hy- 
drostatic resistance of half the vessel. The area of the figure 
c 1 d is manifestly half the area of the parallelogram ace d; and 
us there is a similar figure on the other side of the vessel, the 
total area representing the hydrostatic resistance will be equal to 
half the height of the wave acting against the breadth of the ship. 
Supposing that no disturbing forces were in existence in in- 



416 



STEAM NAVIGATION. 



terfering "with the upward and downward motio.ii of the water, 
a particle of water at the forefoot b, fig. 48, would, as the vessel 
moTed forward, follow the curved line b a; and if on rising 
ahove the line w l it had not to encounter more of the force 
of gravity, it wonld pursue its course along the dotted line a d. 

Fig. 49*. 




As, however, as soon as the particle passes above w l, it has to 
encounter nearly the whole force of gravity, its momentum will 
not suffice to carry it up far, and it will proceed above the wa- 
ter level only to some such point as a, and will then immedi- 
ately pass downward and astern in the track of the curved hue 
V c. The whole of the ascending and descending particles will 
pursue courses nearly parallel to these tracks ; and such lines 
might be drawn mechanically by a tracing point attached to a 



HOW TO REDUCE LOSS OP MOMENTUM. 



417 



pendulum in the manner already described, only that the half 
of the heat answering to the motion of the particle above the 
water-line, would be reduced in length by the ball being made 
in this part of its motion to compress a spring representing the 
increased power of gravity to which the particle is subjected 
during this part of its course. 

Hitherto we have discovered no source of loss of mechanical 
power in the movement of the water by a vessel passing through 
it, except that involved by the necessity of overcoming a con- 
stant hydrostatic resistance in consequence of the difference in 
the level of the water at the bow and stern. There will, how- 

Fier. 50. 



A 








Tl 






CL 


1 

i 
i 


*1 


\v 


. ■ — 


-~^ ' ^-— 






—■ -"# 




~~ ... i _.— — ~~^ ss 










I 
1 
1 

I 





ever, be the loss of the momentum left in the undulating mass 
of water. But this last loss will be diminished, if we shift the 
midship frame further forward, as say to «, fig. 50, which is one- 
third of the length from the bow, instead of half the length. 
For, although we have still the hydrostatic resistance equal to 
half the height of a above w l multiplied by the breadth of the 
vessel to encounter, yet if the after-body of the vessel be prop- 
erly formed with diverging sides, the undulating mass of water 
will have surrendered most of its power to the vessel in aid of 
her propulsion before it leaves the stern at y. If we suppose 
the vessel to be cut off at the water line, we shall get rid of 
the question of the hydrostatic resistance, as the water rising 
above the water-level will in such case run over the deck ; but 
the momentum of the undulating mass will remain, and the ob- 
ject to be attained is so to form the stern part of the vessel that 
the npward motion of the water above the water-line at the 
stern shall be resisted, whereby the mechanical power resident 
18* 



418 STEAM NAYIGATION. 

in the heaving water will he communicated to the vessel. This 
is done at present practically by causing the stern part of the 
vessel to spread outwards near the load water-line, so that tho 
ascending column of water is intercepted by it and gradually 
brought to rest. 

The rise of water at the how, it will he observed, increases 
not merely the hydrostatic pressure against which the vessel 
has to force her way, hut also the opposing area against which 
the pressure acts. In like manner the deficient height of water 
at the stern diminishes both the pressure and the pressed area. 
It is very important, therefore, that the difference of level at 
the how and stern should be as small as possible. And although 
we have supposed that the height of the w T ave a, fig. 50, would 
only be the same if we shifted forward the centre frame, it 
would in point of fact be higher if the same speed of vessel 
were maintained. On this ground, therefore, it appears prefera- 
ble to maintain the midship frame near the position shown in 
fig. 48, the more especially as the forward and ascending cur- 
rent due to the friction of the bottom of the vessel on the water 
has a tendency to bring the surface of the water relatively with 
the ship into the condition represented by the waving-line 
x t asy. Before entering upon the consideration of the friction 
of the bottom, however, it may be stated that the hydrostatic 
resistance consequent on the increased elevation beginning at w 
instead of at x is not all loss. Tor while the height of the wave 
increases the pressure of the water beneath, it also helps to sep- 
arate the water ; and if the vessel be made without any straight 
part between the fore and after-bodies, a portion of the increased 
elevation which the mean water-line w l receives at the bow, 
will be retained to increase the elevation of the water at the 
stern, so that under certain conditions nearly the whole of the 
power expended in moving the water would be theoretically re- 
coverable. In practice, however, such a result is never reached ; 
and however perfect the arrangements for recovering the power 
may be made, yet a certain percentage of it is lost at every 
step ; and the safest indication is to employ such a form of vessel 
as will disturb the water as little as possible. This will be a 
body of the form which I have indicated with a considerable 



BEST MODE OF SHAPING VESSELS. 419 

proportion of length to breadth, so that the vessel may be sharp 
at the ends. A length of 7 times the breadth is found to be a 
good proportion for such speeds as 15 or 16 miles an hour. But 
the proportionate length that is advisable, will increase with the 
intended speed. 

It is not difficult when the intended speed of the vessel and 
also its length and breadth are determined, to find what the 
proper form of the vessel will be, and also the height of tho 
wave which the vessel will raise at the midship frame by her 
passage through the water, one-half, of which height multiplied 
by the breadth of the vessel will be the measure of the hydro- 
static resistance. For as each particle of water at the stern has 
to describe the motion described by the ball of a pendulum 
which makes a double beat during the time that the vessel 
passes through her own length, the breadth of the arc will an- 
swer to half the breadth of the vessel, and the vertical height of 
the arc or the vertical distance fallen by the ball in passing from 
the highest to the lowest part of the arc, will be the height of the 
wave raised at the midship frame — that being the height neces- 
sary to give the velocity of motion, with which the particles of 
water must be moved sideways through half the breadth of the 
vessel, to enable the vessel to pass through in the prescribed 
time. If we suppose the ball of the pendulum to be replaced by 
a mass of liquid moving in a circular arc, the motion of this 
liquid will be the same — except in so far as it is affected by 
friction — as if it were frozen and suspended by a rod of the 
same radius as the arc ; but if the mass of liquid be large so as 
to occupy any considerable part of the length of the arc, the 
motion will not be the same as that of a suspended point, as the 
whole of the particles will no longer rise and fall through the 
same height, while all of them will have still to be moved with 
the same velocity. So also if we have a tube open and turned 
up at both ends, and if we pour water into it and depress the 
water in one leg so as to disturb the equilibrium, the water 
when released will vibrate upward and downward like a pendu- 
lum. Such a tube is represented in fig. 51, where e a b n is the 
tube which is filled with water to the level of x. If the level in 
one leg be depressed from c to g, it will rise in the other leg 



!E_ 



X 



420 STEAM NAVIGATION. 

from d ton; and if the depressing force be now withdrawn, 

the water will fall from h with a velocity corresponding to ita 

height above g, and will be carried by its momentum above o 

to e, iust as the ball of a pendulum ascends in 
Fio\ 51. J , , 

T t -j its arc by the momentum it possesses — and the 

water will continue to oscillate up and down 
H like the ball of a pendulum, until it is finally 
d brought to rest by friction. If the tube be of 
equal bore throughout and be bisected in o, 
then as the accelerating force is the difference 
in the masses of the two unequal columns di- 
vided by their sum, the accelerating force will 
be represented by e g divided by o a b d, or what is the same 
thing, by e abf ; or it will be proportional to the half of this, 
or to e o divided by c a o. The time of the oscillation or the 
time in which the surface of the water will fall from the highest 
to the lowest point, is equal to that in which a pendulum of the 
length c a o makes one vibration. Hence the time in which the 
surface will pass from the highest point to the lowest, and to 
the highest again, will be that in which a pendulum of the 
length o a o will make two vibrations, or it will be that in 
which a pendulum of four times that length makes one vibra- 
tion, or a centrifugal pendulum of the height equal to o a o 
makes one revolution. These relations equally hold, if we sup- 
pose the same kind of motion which exists in the water to be 
produced by a piston at o ; and the side of the ship may be 
supposed to be such a piston, and if properly formed, the ship 
will impart sideways to the water precisely the same kind of 
motion which exists in the case here illustrated. 

If a sheet of paper be drawn vertically behind a pendulum 

furnished with a tracing point, then 
Fig- 52. if the pendulum be stationary, the 

(2.) tracing point will draw a straight 

line represented by the dotted line 
fig. 52. But if the pendulum be 
put into motion, then the tracer 
will describe the waving line abcd 



SHARPNESS SHOULD VARY WITH SPEED. 



421 



Fig. 53. 



where the point A answers to the stem of a ship, the point 
b to the midship frame, and the point o to the stern ; and the 
paper will pass from a to o during the time the pendulum makes 
two oscillations. Since the pendulum has to make two oscilla- 
tions while the vessel passes through a distance equal to her 
own length, the combined motions of the tracer and pencil will 
delineate the proper form for the side of the vessel ; and if made 
in this form the particles of "water wall have the same motion as 
the ball of a pendulum, which motion enables the water to be 
moved with the minimum of loss. It will be useful, however, 
to take a particular case to show in what manner the proper 
form may be practically determined. 

Suppose a o, fig. 53, to represent the keel of a vessel — which 
we may take at 200 feet long and 40 feet wide 
— and which is intended to maintain a speed 
of 10 statute miles per hour, or 880 feet per 
minute. Now as the vessel has to pass 
through her length, or from a to o, during the 
time that the pendulum p makes a double 
beat, or to pass from a to b, which is 100 feet, 
during the time the pendulum make a single 
beat, there will be 880 divided by 100, or 8*8 
vibrations of the pendulum per minute ; and 
the rod of the pendulum must be of such 
]ength as to produce that number of vibra- 
tions. Now to determine the length of the 
rod of a pendulum which shall perform any 
given number of vibrations per minute, we 
divide the constant number 375*36 by the 
number of vibrations per minute, and the 
square of the quotient is the length in inches. 
Hence 375*36 divided ~bj 8-8 = 42*6, the square 
of which is 1814*76 inches or 151*23 feet, and a pendulum 151*23 
feet long beating in an arc 20 feet long with the paper travelling 
at a speed of 880 feet rer minute, will describe the line a b o, 
which will be the proper water-line for the side of a ship if the 
cross-section be rectangular; and whatever the form of r/oss- 




aH 



4*22 STEA1I NAVIGATION. 

section tin's figure will equally determine the proper area of 
cross-section at each successive frame. If instead of moving at 
10 miles an hour, the vessel has only to move at the rate of 5 
miles an hour, the figure described will be that repres ;:::ed by 
d a e, and the breadths 5 £ in the longer figure and o' V in the 
shorter are the same, both being equal to half the breadth at a a. 
The rod of the pendulum p p passes through the point 5, and the 
pendulum vibrates from the plane of the keel to the plane of the 
eidej so that the chord of the arc in which the vibration is \ 
formed is equal to half the breadth of the vessel, while the 
versed sine or height through which the pendulum falls at each 
beat, will be equal to the height of the wave at the midship 
frame. To find the versed sine of the arc. we divide the square 
of half the chord by twice the length of the pendulum. The 
chord being 20 feet the half of it is 10 feet : and the pendulum 
: Bing 151*23 feet long the double of it is 1 13-46 feet, and 100 
divided by 302*46 = *33 feet or 3*96 inches. The height of the 
wave at the midship frame, in a vessel formed in the manner in- 
dicated, will accordingly be 3*96 inches. : : *af her Bos would be 
the height if the water were moved without friction, so : 
practically the height will be somewhat greater than is here in- 
dicated. 

If we increase the speed of the vessel, or increase the breadth, 
the hydrostatic resistance will increase very rapidly. Thus, if 
the speed of the vessel be increased to 20 miles an hour, or 1,760 
feet per minute, the pendulum will require to make 17" 6 beats 
[ .-;• minute, and its length will be I ". •■; divided by 17*6 == 21 '3, 
the square of which is 453*69 inches. ox i" r feet Now, 100 
divided by 37'8 = 2*6 feet, which will be the height of the wat 
at the midship frame in this case, and the hydrostatic pre— ; 
will be the half of this, or equivalent to 1*3 feet of water acting 
on the breadth of the vessel. In like manner, successive addi- 
tions to the breath of the vessel without increasing the length 
add rapidly to the hydros:;.:; : r distance, as they involve the ne- 
fcyof the ; filiating particles ascending higher and higher in 
t:;; arc to enable the vessel to pas& 



FRICTION OF WATER 423 

FRICTION OF WATER. 

It remains to consider the friction of water upon the bottom 
of the vessel, and this is by much the most important part of 
the resistance which ships have to encounter. Beaufoy made a 
number of experiments to ascertain the amount of this resistance 
by drawing a long and a short plank through the water : and, by 
taking the difference of their resistances and the difference of 
their surfaces, he concluded that the friction per square foot of 
plank was, at one nautical mile per hour, *014 lbs.; at two 
nautical miles per hour, -0472 lbs. ; at three, -0948 lbs. ; four, 
•153 lbs. ; five, '2264 lbs. ; six, -3086 lbs. ; seven, '4002 lbs. ; and 
eight, "5008 lbs. At two nautical miles an hour, the force re- 
quired to overcome the friction was found to vary as the 1*825 
power of the velocity, and at eight nautical miles an hour as the 
1'713 power. Other experimentalists have deduced the amount 
of friction from the diminished discharge of water flowing 
through pipes. If there were no friction in a pipe, the velocity 
of the issuing water should be equal to the ultimate velocity of 
a body falling by gravity from the level of the head to the level 
of the orifice.* But as the velocity is found by the diminished 
discharge to be only that due to a much smaller height, the dif- 
ference is set down as the measure of the power consumed by 
friction. . This mode of estimating the friction is not applicable 
to the determination of the friction of a ship ; for, in the first 
place, the discharge is a measure not of the maximum, but of 
the mean velocity ; and, in the second place, there is every reason 
to believe that the friction per square foot on the bottom of the 
ship is quite different near the bow from what it is near the 
stern. As the water adheres to the bottom there will be a film 
of water in contact with the ship, which will be gradually pat 

* There is sometimes misconception on this subject, arising from a neglect of the 
difference between the ultimate and mean velocities of a falling body. Thus, if 
water flows from a small hole in the side of a cistern, the water will issue with the 
ultimate velocity which a heavy body would acquire by falling from the level of the 
head to the level of the orifice, which, if the height be 16 T \ feet, will be 32, 1 , feet 
per second. The mean velocity of falling, however, is only I6J5 feet per second, 
so that the ultimate velocity is twice the mean velocity. 



424 STEAM NAVIGATION. 

into motion by the friction; and the longer the vessel is the less 
will be the friction upon a square foot of surface at the stern- 
seeing that such square foot of surface has not to encounter sta- 
tionary water, but water which is moving with a certain velocity in 
the direction of the vessel. The film of water moving with the ves- 
sel will become thicker and thicker as it passes towards the stern, 
and it will rise towards the surface by reason of the virtual re- 
duction of weight consequent upon the motion. The whole of 
the power, therefore, expended in friction is not lost, as the 
power expended in the front part of the vessel will reduce the 
friction of the after part ; added to which, the rising current 
which the friction produces may be made to aid the progress 
of the ship, if we give to the after-body of the ship such a con- 
figuration as to be propelled onward by this rising current. 
Finally, when the screw is the propelling instrument, the slip 
of the screw will be reduced, and may even in some cases be 
rendered negative, by the circumstance of the screw working in 
this current ; and whatever brings this current to rest will use 
up the power in it, and so far recover the power which has been 
expended in overcoming the friction. 

In my investigations respecting the physical phenomena of 
the river Indus in India, I observed that the water not only ran 
faster in the middle of the stream, but that it also stood higher 
in the middle, so that a transverse section of the river would 
exhibit the surface as a convex line. At the centre of the river 
the stream is very rapid, but it is slow at the sides, so that boats 
ascending the river keep as close as possible to either bank ; and 
in some parts at the side there is an ascending current forming 
an eddy. I further observed, that not merely were there rapid 
and considerable changes in the velocity, which I imputed 
partly to the agency of the wind in deflecting the most rapid 
part of the current to the one side or the other of the river, but 
there were also diurnal tides ; or, in other words, the stream ran 
more swiftly in the afternoon than in the early morning. This 
had been long before observed, and was imputed to the heat of 
the sun melting the snows in the mountains more during the day 
than during the night. But although such an effect might be 



INFLUENCE OF HEAT ON VELOCITY OF KIYEES. 425 

observable in a single feeder, the river is supplied from so many 
sources at different distances that such intermittent accessions 
would equalise one another. Moreover, the effect of the sun in 
the daytime in swelling the volume of the river, if acting withoul 
any equalising influence, could only produce a wave like a tida; 
wave in the river ; and the increase of velocity would at some 
points take place at night and at some in the morning, whereas 
I found it to take place everywhere at the same time. I finally 
came to the conclusion that the phenomenon is caused by the in- 
fluence of the sun in heating the water of the river, and thereby 
increasing its liquidity and its velocity throughout the whole 
length of the river. The temperature of the water in the river 
is commonly about 94° Fahr., but as the river is wide and shal- 
low, it is rapidly heated and cooled, and there are several de- 
grees difference between the temperature of the day and the 
night. In the early morning the river is coldest, and at that 
time also — other things being equal — its velocity is least. It 
may hence be concluded that any thing which gives more mo- 
bility to the particles of the water in which a vessel floats will 
diminish the friction of the bottom ; and this end seems likely to 
be attained by the injection of air into the water at the stem 
and forefoot or front part of the keel. 

It is not difficult to understand how it comes that the water 
in a river should stand higher at the middle than at the sides, as 
shown in fig. 54. If we hang a weight upon a spring balance 
we shall find the amount of the weight to 
be indicated on the scale or index ; and 
this weight will continue to be shown so 
long as we hold the spring balance sta- 
tionary. But if we allow it to move tow- 
ards the earth with the velocity which 

a heavy body would acquire in falling by gravity, the index of 
the spring will show no tension at all — proving that with this 
amount of downward motion the body imparts no weight. If 
the spring is moved downward slower than a body falls by grav- 
ity, the spring will show that it is sustaining some weight ; but 
at any velocity downward there will be a diminution in the 




i26 



STEAM NAVIGATION. 



Fis:. 55. 



weight of the body answerable to that velocity. *In two columns 
of water, therefore, moving at different velocities, the slower 
will exert most hydrostatic pressure on the pipe or channel con- 
taining it ; and where two such columns are connected together 
sideways, as in a river, the faster must rise to a greater height to 
je in hydrostatic equilibrium sideways with the slower. The 
surface of the water consequently becomes convex, as shown at 
m in fig. 54, where h is the water and abcd the bed. 

It will be seen from these observations that there is a hy- 
draulic as well as a hydrostatic head of water; and the hydraulic 

head is equal to the hydrostatic head, 
diminished by the height due to the 
velocity with which the water flows. 
This law is further illustrated by fig. 
55, which represents a bulging vessel 
in which the water is maintained at a 
uniform height by water flowing into 
it at the top, while it runs out at e at 
the bottom. The velocity with which 
the water flows downward from a to 
e, varies with the amount of enlarge- 
ment or contraction of the vessel ; 
and the height of water which will 
be supported in the small pipes 5, c 
and d, varies as the velocity of the water at their several points 
of insertion. Thus, the area at b, being greater than the area 
at a, the velocity will be less, and consequently the water will 
stand in the small pipe 5 at a point higher than the surface of a. 
The area at d being less than the area at a, the velocity will be 
greater ; and the height of the water in the small tube d will not 
come up to the level of a. At c, the velocity of the water being 
very great^ not only no height of column will be supported in the 
tube c there inserted, but the water will be sucked up through 
the inverted tube c, out of the small cistern f ; and if there bo 
no cistern air will be drawn through the tube. So also in fig. 
56, if a pipe be led out at the bottom of a cistern of water, a 




FRICTION OF THE BOTTOM OF VESSELS. 



427 




A 



fig. 56. hole bored in any part of the pipe will draw air and 
not leak water, so long as the water is running out 
of the bottom of the pipe. 

It follows, from these considerations, that the 
stratum of water put into motion by the friction of 
the vessel will rise to a higher level than the sur- 
rounding water, which is at rest; and advantage 
should be taken of this ascending current to aid in. 
propelling the vessel, by spreading out the stern part 
so as to intercept and derive motion from the rising 
water. This is, to some extent, done in common ves- 
l|j{ sels by the greater breadth which is given to the 
stern part near the water level; and although no 
very tangible reason is commonly adduced for the practice be- 
yond that of affording greater accommodation for the cabins, the 
method of expanding the breadth at the stern is also useful in 
utilising the ascending current. The manner in which the ship 
acts upon the water in urging it into motion by friction is not 
known. But it is known that the vessel carries a film of water 
with it in the same manner as the belt-pump ; and it is known 
that the particles of water nearest the vessel move with a velocity 
nearly the same as that of the vessel, and that the motion of each 
particle diminishes in amount the further it is from the vessel, 
until those particles are reached which are wholly at rest. The 
moving film may consequently be regarded as a roller interposed 
jetween the bottom of the vessel and the water ; and such a 
roller would enable the vessel to move forward with twice the 
speed that the roller itself moves at. But before this roller can 
be set into motion, there will be a good deal of slip or pure fric- 
tion, just as there is in the driving-wheel of a locomotive in 
starting the train. It is not known what length of vessel will 
suffice to move the film of water with the maximum velocity it 
can attain with any given speed of the ship ; nor is it known 
what the maximum speed of the film is with any given velocity 
of the ship. The speed will always be less than the speed of the 
ship, but how much less is not known ; and this speed, when 
once attained, will not be increased, as when it is reached the 



428 STEAM NAVIGATION. 

power communicated by the friction of the bottom will be 
balanced by the power consumed in maintaining the motion 
among the internal particles. Up to a certain point, therefore, 
the friction upon a square foot of the ship's bottom will diminish 
with the distance from the stem ; and the thickness of the moving 
film will also increase with that distance. But when that point 
of the length has been reached, the friction per square foot will 
become uniform, and there will be no further increase in the 
thickness of the film. 

Instead, however, of supposing the film interposed between 
the stationary water and the moving bottom to be a single 
roller, it will be a nearer approximation to the truth if we sup- 
pose it to be composed of an infinite number of rollers, a a a a 
in fig. 57, where we may suppose s s to be the ship, while the 
line extending from roller to roller represents the 
^ amount of motion which the water receives from 

S each successive length of the ship, and which dimin- 

'V? ishes as we recede from the stem until we reach the 
point a b, where the pure friction of the bottom 
upon the particles balances the power consumed in 
maintaining the internal motion of the water, and 
which power is ultimately transformed into heat. 
The whole power concerned in propelling the ves- 
sel is consumed either in moving the water or in 

- heating it. The greater part of the power expended 

- in moving the water aside at the bow, is recovered 
by the closing of the water at the stern ; and most 
of that expended in friction in producing a rising 

current is recoverable by giving a proper configuration to the 
stern. Of the heat generated, the whole is not lost, as it will 
give greater mobility to the particles of the water, which will 
also be given by heating the bottom, as has been done in some 
steam-vessels, by converting the bottom into a refrigeratiDg sur- 
face for condensing the steam ; and by which arrangement the 
bottom itself has been heated to some extent. On the whole, 
however, that arrangement will be the most advantageous for 
reducing resistance by which the least motion is given to the 



A° 



SPEED OF VESSELS OF A GFVEN POWER. 12.9 

water, and the least heat generated in it ; and the smoothness 
of the rubbing surface will somewhat affect that question. In 
pipes, it has been found that there is no increase of friction from 
increase of pressure. But it must not be therefore inferred that 
in vessels the friction per square foot is precisely the same 
at every point in the depth, any more than at every point of 
the length ; for the moving water has to escape to the surface, 
and the dimculty of the escape will be the greater tbe further 
the surface is off. If we knew the ratio in which, the resistance 
of a vessel increased with the length and with the depth, we 
should be able to tell what form the vessel should have, in order 
to offer the least resistance. But it is quite certain that the re- 
sistance per square foot of the bottom does diminish with the 
length in some proportion or other ; and as the resistance also 
diminishes as the wetted perimeter, and as relatively with the 
sectional area, the wetted perimeter of large vessels is less than 
that of small, it is easy to understand how it comes that large 
vessels are swifter than small with the same proportion of pro- 
pelling power. If we double the breadth and immersed depth 
of a vessel, we double the length of its perimeter. But we in- 
crease its sectional area fourfold ; and as with any given length, 
and with equally fine ends, the wetted perimeter is the measure 
of the resistance, it follows that the large vessel will require less 
power per ton or per square foot of immersed section to main- 
tain any given speed. 

SPEED OF STEAM VESSELS OF A GIVEN POWER. 

There were no accepted rules for ascertaining the speed that 
a steam vessel of a given type would probably obtain with en- 
gines of a given power, until the appearance of the first edition 
of my Catechism of the Steam-Engine, when I published the rule 
which had long been employed by Messrs, Boulton and Watt 
for determining this point. This rule was founded on a long- 
continued series of experiments on steam vessels of different 
types ; and for similar hinds of vessels the results it gives have 
been found very nearly to accord with those subsequently ob- 



430 STEAM NAVIGATION. 

tained by experiment. This rule, "which proceeds on the suppo- 
sition that the engine power required for the propulsion of a 
vessel varies as the area of the immersed midship-section, and aa 
the cube of the speed, has been already referred to in page 77 
as an example of the application of equations, and in algebraical 
language it is as follows : — 

If s be the speed of the vessel in knots per hour, a the area 
of the immersed midship section in square feet, c a numerical 
coefficient, varying with the form of vessel and to be fixed by 
experiment, and p the indicated horse-power : then 

s 3 a, s 3 a ^ , PC 

p = — - c = — and s = Jy — 

C P V A 

In words these rules are as follows : — 



TO DETERMINE THE POWEE XECES8AEY TO EEALISE A GIVEX SPEED 
IX A STEAM VESSEL BY BOEXTOX AXD WATT'S EEXE. 

Rexe. — Multiply the cube of the given speed by the area in 
square feet of that part of the midship section of the vessel 
lying below the uaier-line, and divide the product by a cer- 
tain coefficient of which there is a different one for each 
particular type of vessel. The quotient is the indicated 
power in horses that will be required to give the intended 
speed. 

Example. — The steamer ' Fairy,' with an immersed sectional 
area of TL§ square feet, and a coefficient of 465, attained on 
trial a speed of 13*3 knots per hour. What indicated power 
must have been exerted to attain this speed ? 

Here the cube of 13*3 is 2352*637, which multiplied by 71*5 
= 168210*9, and this divided by -465 is equal to 363 horse-power, 
which was the power actually exerted in this case. 

In the first edition of my Catechism of the Steam-Engine- 
the coefficients of a number of steam- vessels were given, which 
had been ascertained experimentally by Boulton and "Watt; and 
in &«£ first edition of my Treatise on the Screw Propeller, pub- 
lished in 1 852, I recapitulated a number of the coefficients of 



■RULES FOR FINDING SPEED OF VESSELS. 431 

the screw steamers of the navy, which had then been recently 
ascertained by the steam department of the navy, as also the co- 
efficients obtained by multiplying the cnbe of the speed — not by 
the area of the midship section, but by the cube root of the 
square of the displacement — and dividing by the indicated power. 
The displacement of the 'Fairy ' at the trial, at which the speed 
was 13*3 knots, was 168 tons. Now the square of 168 is 28224,. 
the cube root of which is 30*45 nearly, and this multiplied by the 
cube of the speed 2352*637 and divided by the indicated power, 
363 horses, gives 197 as the coefficient proper to be employed 
when this measure of the resistance is adopted. Neither the 
immersed section, however, nor the displacement, is the proper 
measure of the resistance in steam vessels ; and I pointed this 
out in the first edition of my Treatise on the Screw Propeller, in 
1852, and suggested the wetted perimeter as a preferable meas- 
ure of the resistance ; the perimeter being a measure of the 
friction ; and nearly the whole of the resistance of well-formed 
ships being produced by friction. Under this view the velocity 
of ships with any given perimeter and propelling power would 
fall to be considered in much the same way as the velocity of the 
water flowing in rivers or canals, and in which the speed with 
any given declivity of the bed varies as the hydraulic mean 
depth, or in other words as the sectional area of the stream di- 
vided by the wetted perimeter. In such a comparison the en- 
gine power of the ship answers to the gravitation of the stream 
down the inclined plane of the bed, while the area of the trans- 
verse section of the ship beneath the water-line divided by the 
wetted perimeter constitutes the hydraulic mean depth of the 
ship. This measure of the resistance, however, though accurate 
enough for short vessels, is not applicable to long vessels without 
some allowance being made for the inferior resistance of long 
vessels of the same sharpness at the ends, in consequence of the 
proportion of power which long vessels recover, especially if 
propelled by the screw or any other propeller situated at the 
stern. 



432 STEAM NAVIGATION. 

TO DETEEMINE BOULTON AND WATT'S COEFFICIENT FOE ANY GIVEN 
TES3EL OF WHICH THE PEEFOEMANCE IS KNOWN. 

Rule. — Multiply the cube of the speed, in knots per hour by the 
area in square feet of the immersed transverse section of the 
vessel, and divide the product by the indicated horsepower. 
The quotient icill be the coefficient of that particular type 
of vessel. 

Example. — The steamer Fairy, with, an area of immersed 
section of 71^- square feet, and 363 indicated horse-power, at- 
tained a speed of 13'3 knots an hour. What is the coefficient 
of that vessel? 

Here 13'3 cubed = 2352'637, which multiplied by Tl'5 and 
divided by 363 horse-power = 465, which is the coefficient of 
this vessel according to Boulton and "Watt's rule. A good num- 
ber of coefficients for different vessels is given at page 77. 

to determine what speed well be attained bt a steam vessel 
of a given type with a given amount of engine powee, by 
bout-ton and wattes etjle. 

Rule. — Multiply the indicated horse-power by the coefficient 
proper for that particular type of vessel, and divide the 
product by the area of the immersed transverse section in 
square feet. Extract the cube root of the quotient, which 
will be the speed that will be obtained in knots per hour. 

Example. — What speed will be obtained in a steamer of 
which the coefficient is 465, and which has an immersed section 
of 71 J square feet, and is propelled by engines exerting 363 
horse-power. 

Here 363 x 465=168795, which divided by 71-5=2360. The 
logarithm of this is 3*372912, which divided by 3=1-124304, 
the natural number answering to which is 1331. !Now the in- 
dex of the divided logarithm being 1, there will be two integers 
in the natural number answering to it, which will consequently 
be 13-31, and this will be the speed of the vessel in knots -per 
hour. 



MEAN VELOCITY OF WATER IN CANALS, ETC. 433 

The coefficient of a steamer sometimes varies with the speed 
with which the vessel is propelled. If the vessel is properly 
formed for the speed at which she is driven, then her coefficient 
will not "become greater at a lower speed ; and if it becomes 
greater, the circumstance shows that the vessel is too bhmt. 
"When the ' Fairy' was sunk to a draught of 5 feet 10 inches, her 
speed was reduced to 11*89 knots, and her coefficient was re- 
duced from 465 to 429, showing that she worked more advan- 
tageously at the higher speed and lighter draught. The ' War- 
rior,' which when exerting 5,469 horse-power attained a speed 
of 14.356 knots with a coefficient of 659, attained when exerting 
2,867 horse-power a speed of 12*174 knots with a coefficient of 
767; and when exerting 1,988 horse-power a speed of 11"040 
knots with a coefficient of 825. This shows that the '"Warrior' 
is too blunt a vessel for a high rate of speed. 

It will be satisfactory to ascertain the comparative eligibility 
cf the forms of the 'Fairy' and the ""Warrior,' which we may 
easily do by comparing the speed attained by each, with the 
rpeed wliich would be attained by an equal weight of water 
running in a river or canal, and impelled by an equal motive 
force. The rule for determining the speed of water flowing in 
rivers or canals of any given declivity is as follows : — 

TO DETERMINE THE MEAN VELOCITY WITH WHICH WATEE WILL 
FLOW THROUGH CANALS, AETEEIAL DEALXS, OE PIPES, EUNNTXG 
PAETLY OE WHOLLY PILLED. 

Rule. — Multiply the hydraulic mean depth in feet ~by twice the 
fall in feet per mile. Extract the square root of the product, 
which is the mean velocity of the stream in feet per minute. 

Now the 'Fairy,' when realizing a speed of 13*3 knots per 
hour with 363 horse-power, had a draught of water of 4*8 feet ; 
a sectional area of 71 '5 feet; a wetted perimeter of 24*7 feet, 
and a displacement of 168 tons. The hydraulic mean depth be- 
ing the sectional area in square feet, divided by the length of 
the wetted perimeter in feet, the hydraulic mean depth will in 
this case be 71*5, divided by 24*7=2-9. 

The engine made 51 - 6 revolutions per miuute, and the screw 
19 



431 STEA3I NAVIGATION 

258 revolutions per minute, "being fire times the nnmber of resv« 
olntions of the engines. The stroke is 3 feeu and the pitch of 
the screw 8 feet. 

Xow a horse-power being 33,000 lbs., raised 1 foot per min- 
nte, and as there were 363 horse-povrer exerted, the total effort 
of the engines will be 363 times 33,000, or 11,979,000 lbs., raised 
throngh 1 foot each minute. Bnt the engine makes 51 "6 revolu- 
done each minute, and the length of the double stroke is 6 feet^ 
so that the piston moves throngh 309 '6 feet per minnte; and 
the power being the product of the velocity and the pressure, 
the power 11. i ". .'. ) lbs. divided by the velocity of the piston, 
309*6 feet per minute, will give the mean pressure urging the 
pistons, which will be 38,691 lbs. But the speed of the screw- 
shaft being five times greater than that of the engine-shaft., the 
sure urging it into revolution must, in order that there may 
be an equality of power in each, be five times less ; or it will be 
7.! ■! 5 lbs. moving throngh 6 feet at each revolution. Then the 
pitch of the screw being 8 feet, the thrust of the screw will be 
less tjian 7.538, in the proportion in which 6 is less than 8, or it 
will be 5,653, supposing that there is no loss of power by slip 
and friction. It is found on an average in practice, that about 
one-third of the power is lost in slip and friction ; and the actual 
thrust of the screw-shaft will be about one-third less than the 
theoretical thrust, or in this case it will be 3,769 lbs. or 1*68 
ton. Xow. in order that 168 tons of water may gravitate down 
an inclined channel with a weight of 1'68 ton, the declivity of 
the channel must be 1 in 100. In 1 mile, therefore, it will be 
52*80 feet. A cubic foot of salt-water weighs 61 lbs., so that 
there are 35 cubic feet in the ton, and in 168 tons there are 
5,880 cubic feet. Dividing this by the sectional area 71 '5 feet, 
we get a block of water 82*2 feet long, and with a cross-section 
of 71*5 feet, weighing 168 tons; and the wetted perimeter being 
£-7 feet, and the length S2*2 feet, we get a rubbing area of 
2020*34 feeV; and as the friction on this surface balances the 
weight of 3,769 lbs., there will be a friction of 1*8 lb. on 
square foot. If this block o&:water be supposed to be let down 
a channel falling 1 in 100, its t :-'_ ; : . Trill go on increasing until 



SHIPS COMPARED WITH RIVERS. 435 

tlio friction balances the gravity, which, according to the rule 
given above, "will be when the water attains a speed of 11 miles 
an hour, from whence we conclude that the sum of the resist- 
ances of a well-formed ship are less than the friction alone of an 
equal weight of water of the same hydraulic depth, moved in a 
pipe or canal by an equal impelling force. If instead of taking 
the declivity in 2 miles, as the rule prescribes, to ascertain the 
velocity of the water, we take the declivity in twice 2, or 4 
miles, we shall arrive at a pretty exact expression of the speed 
of the vessel in this particular case. Taking the knot at 6,101 
feet, 13*3 knots will be equal to 15*3 statute miles, and the de- 
clivity in 1 mile being 52*8 feet, the declivity in 4 miles will be 
211-2 feet. Multiplying this by 2*9, the hydraulic mean depth, 
we get 612*48, the square root of which is 24*7, which multi- 
plied by 55, gives the speed of the water in feet per minute = 
1,358*5. This, multiplied by 60, gives 81,510 feet as the speed 
per hour, and this divided by 5,280, the number of feet in a 
statute mile, gives 15*4 as the speed in statute miles per hour. 

The ' "Warrior,' with a displacement of 8,852 tons, a draught 
of water of 25-J- feet, an immersed midship section of 1,219 square 
feet, and 5,469 horse-power, attained a speed of 14*356 knots, or 
16*6 statute miles. The number of strokes per minute was 34£, 
and the length of the double stroke 8 feet, while the pitch of the 
screw was 30 feet. The wetted perimeter is 88 feet, which 
makes the mean hydraulic depth 13*8 feet. The power being 
5,469 horses, 33,000 times this, or 180,477,000 lbs., will be lifted 
1 foot high per minute. But as the piston travels 54*25 times 8 
feet, or 434 feet each minute, the load upon the pistons will be 
415,845 lbs. The pitch of the screw, however, being 30 feet, 
while the length of a double stroke is 8 feet, the theoretical 
thrust of the screw will be reduced in the proportion in which 
30 exceeds 8, or it will be 110,892 lbs. If from this we tak'j 
one-third, on account of losses from slip and friction, we get 
73,928 lbs., or 33 tons, as the actual thrust of the screw. 

Now 8,852, which is the displacement in tons, divided by 
33 tons, which is the motive force in tons, gives 268, or, in other 
-vords, the declivity of the channel must be 1 in 268, in order 



i36 STEAM NAVIGATION. 

that 8,852 tons may press down the inclined plane with a force 
of 33 tons. This is a declivity of very nearly 20 feet in the 
mile, or 40 feet in two miles, or 80 feet in twice two miles. 
The mean hydraulic depth being 13 '8 feet, 80 times this is 1,104, 
the square root of which is 33'2, which multiplied by 55=1,826 
feet per minute, or multiplying by 60=109,560 feet per hour. 
Dividing by 5,280, we get the speed of 20 miles per hour, which 
ought to be the speed of the ' Warrior ' if her form were as 
eligible as that of the 'Fairy.' The speed falls 3*4 miles an hour 
short of this, which defect must be mainly imputed to the de- 
ficient sharpness of the ends for such a speed and draught, and 
the increased resistance consequent on the greater depth. 

In a paper by Mr. Phipps, on the ' Resistances of Bodies pass- 
ing through Water,' read before the Institution of Civil En- 
gineers in 1864, it was stated that these resistances comprised 
the Plus Eesistance, or that concerned in moving out of the way 
the fluid in advance of the body ; the Minus Resistance, or the 
diminution of the statical pressure behind any body when put 
into a state of motion in a fluid ; and the Prictional Resistance 
of the surface of the body in contact with the water. 

The Plus Eesistance of a plane surface one foot area, moving 
at right angles to itself in sea water, was considered to be 

64*2 x 'B 2 
E = , and the Minus Resistance was one half the Plus 

2<7 ' 

Resistance. 

For planes moving in directions not at right angles to them- 
selves, the theoretical resistances were, for the Plus Pressure — 

r, a -, -r> /S'64'2y 2 

# = -, and R = -, 

r 2 ' 2g ' 

the Minus Pressure being one-half the above ; where E was the 
resistance of the inclined plane; a, the area of the projection 
of the inclined plane upon a plane at right angles to the direc- 
tion of motion; r, the ratio of the areas of the projected and 
the inclined planes ; and S, the area of a square-acting plane 
of equivalent resistance with the inclined plane. 

But, besides these theoretical resistances, the experiments of 
Beaufoy showed, that when the inclined planes were of moderate 



RECENT COMPUTATIONS OF RESISTANCE. 437 

length only, the Plus Resistance was considerably in excess of 
the above ; so that when the slant lengths of the planes were to 
their bases in the proportion of 

2 to 1, 8 to 1, 4 to 1, and 6 to 1, 

the actual resistances exceeded the theoretical, as 

1-1 to 1, 1-88 to 1, 3-24 to 1, and 6"95 to 1. 

Mr. Phipps proposed a method of approximating to these ad- 
ditional resistances, by adding the constant fraction of jth of a 
square foot for every foot in depth of the plane to the quantity 
S previously determined, which empirical method he found to 
agree nearly with the results of Beaufoy's experiments. 

The resistances of curved surfaces, such as the bows of ships, 
were adverted to, the method of treating them being to divide 
the depth of immersion into several horizontal layers, and then 
again into a number of straight portions, and to deal with each 
portion as a separate detached plane, according to the preceding 
rules. 

The question of friction was then considered. The experi- 
ments of Beaufoy were referred to, giving 0*339 lb. per square 
foot as the co-efficient of friction for a plained and painted sur- 
face of fir, moved through the water at 10 feet per second, the 
law of increase being nearly as the squares of the velocities, viz., 
the l - 949 power. Mr. Phipps was, however, of opinion, that a 
surer practical guide for determining the coefficient of friction 
would be, by considering all the data and circumstances of a 
steam-ship of modern construction, moving through the water 
at any given speed. The actual indicated horse-power of the 
engines being given, the slip of the paddles being known, and the 
friction and other losses of power approximated to, it was clear 
that the portion of the power necessary to overcome the resistance 
of the vessel might be easily deduced. By determining approxi- 
mately, by the preceding rules, the amounts of the Plus, the 
Minus, and the Additional Head resistances, and deducting them 
from the total resistance, the remainder would be the resistance 



438 steam xayigatio:h". 

due to tlie friction of the surface. By this process, and taking as 
an example, the iron steam-ship 'Leinster,' when perfectly clean, 
and going on her trial trip 30 feet pe? second in sea-water, her im- 
mersed surface being 13,000 square feet, the coefficient of fric- 
tion came out at 4*34 lbs. per square foot. Beanfoy's coefficient 
of 0*339 lb. per square foot at 10 foot per second would, accor ling 
to the square of the velocities, amount to 3 - 051 lbs. at 30 feet per 
second. The difference between this amount and the above 4'34 
lbs. might be accounted for by a difference in the degree of 
roughness of the surfaces. 

Other methods for the determination of the coefficient of 
friction were then discussed. One. derived from the known 
friction of water running along pipes, or water-courses, was 
shown to be considerably in excess of the truth. It was founded 
upon the observed fact, that at a velocity of 15 feet per second 1 , the 
friction of fresh water on the interior of a pipe was 25 oz.* per 
square foot. Applying this to the ship 'Leinster,' and increasing 
the friction as the square of the velocities up to 30 feet per 
second, the above friction would become 100 oz.. or 6^ lbs., per 
square foot, which, acting upon 13,000 square feet of surface, 
would absorb, at the above speed, no less than 4,395 H.P.. whilst 
the total available power of the engines (after deducting from the 
indicated 4.751 H.P. ^th for friction, working air-pumps, and 
other losses, and ith of the remainder for the observed slip), was 
only 3,421 H.P. ; thus showing an excess of resistance equal to 
974 H.P., without allowing any power to overcome the other re- 
sistances. The assumption of 25 oz. being the proper measure 
of the friction per square foot, at a velocity of 15 feet per second. 
upon the clean surface of an iron ship, seemed to have arisen 
from the opinion, very generally entertained, that there was no 
difference in the amount of friction in pipes and water-course-, 
whether internally smooth like glass, or moderately rough lit : 
cast-iron, and that the surfaces of ships were subject to the same 
action. The comparatively recent experiments, in Frame, of fch< 
late M. Henry Darcy were in opposition to the above view, and 

* For sea water this quantity must be increased as the specific gravity, or aa 
62 5 to G4-2. 



EFFECT OF SMOOTHNESS OF SURFACE. 439 

showed that the condition as to roughness of the interior of a 
pipe modified the friction considerably. Thus, with three differ- 
ent conditions of surface, the coefficients were : 

A. Jron plate covered with bitumen made very smooth, 0*000432 
13. New cast-iron ...... . 0*000584 

O. Oast-iron covered with deposits . . . .0*001167 

The friction was, therefore, nearly as 1, H, and 8. 

As there appeared no reason to doubt the correctness of M. 
Darcy's experiments, even in pipes the notion of the friction being 
uninfluenced by the state of roughness of the interior could no 
longer be entertained. The 25 oz., previously mentioned as the 
measure of friction per square foot for the interior of pipes and 
water-courses, could not, therefore, be regarded as a constant 
quantity, applicable to all kinds of surfaces; but from Mr. 
Phipps' calculations, it appeared to come intermediately between 
the coefficients of the surface B and 0, given in the above scale ; 
as at 15 per second, 

A would give 13 j oz. per square foot 
B " 20 " " 

and O " 40 " . " 

Besides, there was another cause for an excess of friction in 
pipes and water-courses, over that upon ships, even when the 
surfaces were equally smooth. It arose from the circumstance, 
that where the velocity of the water in a pipe, or open water- 
course, was spoken of, the meaning was, its average velocity; 
whilst the velocity of a vessel through still water meant what the 
words implied, namely, the relation of the vessel's motion to the 
fluid at rest. If the case were taken of a water-course of such 
width, that the friction of the bottom only need be considered, 
with an average velocity of flow of 15 feet per second, the friction 
Tipon the bottom would be equal to 25 oz. per square foot ; but 
according to the rules generally used, an average velocity of 15 
feet per second corresponded to a surface velocity of 16*G6 feet 



440 STEAM NAVIGATION. 

per second, which was the velocity witli which a vessel should 
pass through still water, to give an equal friction upon its sides. 
According to Beaufoy, the velocity of 16*66 feet per second would 
produce a friction of '932 lbs. or 14*91 oz., where 15 feet would 
only give 12*2 oz. The difference between 14*91 oz. and 25 oz. 
(equal to 10*09 oz.) must, therefore, Mr. Phipps thought, be set 
down to the different degree of roughness of the surfaces in the 
water-course and the vessel. 

Taking then 4*34 lbs. as the friction per square foot of a new 
iron ship, moving through the water at a speed of 30 feet per 
second, it would be found, Mr. Phipps considered, that this was 
equal to the 2 oT-oi" P ar ^ °f the P ms resistance of a plane 1 foot 
square, moving through the water at right angles to itself at the 
above velocity. Also, as the resistance of both planes increased 
according to the same law of the square of the velocities, the 
ratio of 1 to 207*06 would subsist at all velocities. 

64*2« 2 1 

The ratio was — r to 4*34 lbs. = 



2 g 207*06 

Calling the ratio r, and the whole frictional surface in square feet 
s, and S, as before, the area of a square-acting plane of equiv- 
alent resistance, then 

S = s -T- r = s -*- 207*06. 

As an example of the application of the previous deductions, 
the performance of the steam-ship 'Leinster,' on her trial trip, 
when going through sea- water at a speed of 30 feet per second, 
was referred to. 

In this case — 

?n, the area of the immersed midship section was 336 sq. ft. 

d, the draught of water 13 ft. 

?', the reduced ratio of the slant length of the bow 

to the projection 10 to 1. 

r', the same for the stern . . . . 10 to 1. 
r", the ratio of 1 square foot of square-acting 

plane, to 1 square foot of frictional surface 207*06 to I. 

w, the velocity in feet per second ... 80 



EXAMPLE OF STEAMER ' LEINSTER.' 441 

w, the weight of a cubic foot of sea-water . 64*2 lbs. 

f, the area of the frictional surface . . . 13,000 sq. ft* 

Calling P, the Plus, or head resistance ; Jf, the Minus, or stern 
resistance ; A, the Additional Head resistance ; F, the Friction- 
al, or surface resistance ; S, the area of a square-acting plane 
having an equal resistance with each of the above ; and i?, the 
total resistance ; 

Then, P = ™ =S= ^ = 3-36 sq. ft. 
' r 3 100 ^ 



= S=i S ^= 1-68 



2 /..* 



A= ~ =&= 1*86 

7 

F =™^ = S = 62-78 

207-06 



S= 69-68 



64"2^ 2 
It = 69'68 x -- == 69-68 x 900 = 62,712 lbs 

lbs. 
JET (Eealized Power) = 62,712 x 30 -v- 550 = 3420*66 H.P. 

H' (Gross Power) including the slip and other losses, = 

8420-66 x ^ = 4751 H.P. 

72 

Thus, by ascertaining the value of S for any vessel, which 
was entirely independent of velocity, it would be easy to deter- 
mine the power necessary to propel it at any required speed, or 
the speed being given, to find the corresponding power. 

64-2 V 2 
Generally H=V8 — ■ ~ 550 (1 ) 

Or, because for sea water 64-2 was very nearly equal to 2 g, 



V 3 S 



19* 



442 STEAM NAVIGATION. 

When the slip and other losses were in the same proportion as 
in the ' Leinster ' : 

•*•=*£ . m 

When the gross power was given, and the velocity was required 

m s ' x550 \i 
t r=\ (4) 

Mr. Phipps then proceeded to examine the question of tho 
influence of form in reducing the resistance of vessels. 

It was argued that, in vessels of similar type to the 'Lein- 
ster,' where y^ths of the whole resistance was due to friction, 
and only y^th to considerations involving the question of 'form,' 
no minor modifications of the latter could have much effect in 
diminishing the total resistance. The case of other vessels of 
different type, more bluff in the bows and not so fine in the run, 
was adverted to, and a particular instance was discussed, where 
the inertial resistance was supposed to be equal to ith of the 
total resistance, aud the slant length of the bows to the base to 
be as 6 to 1. If such a vessel were altered, so as to make the 
above proportion 8£ to 1, the improvement would only diminish 
the total resistance by T Vfch. 

The conclusion that the friction of ships constitutes the 
largest part of their resistances, was first pressed upon me in 
1854, in which year I built two steamers with water-lines formed 
on the principle of imparting to the particles of water the mo- 
tion of a pendulum, as already explained. I found, as I expected, 
that these vessels passed through the water with great smooth- 
ness, and without in any measure raising the water in a wave at 
the bow, as was a common practice in the older class of steam- 
boats. ISTevertheless I did not obtain a speed much superior to 

* If for fresh water W x 0-97 = Gross H.P. 
t If for fresh water F-*- 0"99 = Velocity. 



MODES OF PREDICTING VELOCITY. 443 

that of vessels less artistically formed ; and the conclusion be* 
came inevitable — seeing that all other known causes of resist- 
ance had been reduced to a minimum without material benefit 
to the speed — that the friction, which alone remained unchanged, 
must constitute the main element of resistance ; and other things 
being alike, the friction of a vessel, as of a river, would, in such 
case, be measurable by the wetted perimeter of the cross-section. 
It was further plain, that as there was not much difference be- 
tween the resistance of a vessel formed with pendulum or wave 
curves, and that of well-formed vessels of the ordinary configur- 
ation, any mode of computing the resistance applicable in the 
one case would also be applicable without material error in the 
other. These conclusions, which I published in my ' Catechism 
of the Steam-Engine,' in 1856, are now very generally accepted ; 
and when, in 1857, Mi*. Eankine had to compute the probablo 
speed of an intended vessel, he proceeded on the supposition 
that the resistance was due almost wholly to friction, and that 
the friction of a riband of the form of a trochoid or rolling 
wave, of the length of the ship and of the breadth of the wet- 
ted perimeter, would be an accurate measure of the resistance, 
the trochoid being the same order of curve as that which would 
be described by a pendulum. Since, however, a wave moves in 
different parts with different velocities, Mr. Eankine concluded 
that it would be proper to take this circumstance into account, 
and he therefore, instead of taking the actual surface of the ves- 
sel, took a surface so much larger, that its friction would pro- 
duce a resistance equivalent to the increased friction caused by 
the varying velocities of the wave, and the hydrostatic pressure 
consequent upon the difference of level at the bow and stern, 
and which in a well-formed vessel is very small. This additional 
or hypothetical surface Mr. Eankine terms augmented surface ; 
and by using this theoretical surface in his computations instead 
of the actual wetted surface of the ship, he deduces results sin- 
gularly conformable to those obtained by actual experiment. 
The amount of the augmented surface will vary with the sharp- 
ness of the vessel — sharp vessels having the least augmentation; 



iU 



STEAM NAVIGATION. 



and the sharpness is measured "by the sines* of the angles of the 
water-lines at the bow and stem. I shall here introduce Mr. 
Eankine's able investigation, to which the only exception that 
can be taken, so far as I see, is that the resistance per square 
foot produced by friction in every part of the length of the ves- 
sel is not the same, but is more at the fore part, in consequence 
of the necessity of putting the water into motion ; but after this 
has been done, the friction per square foot of the further length 
of the vessel will be uniform. 

The Resistance due to Frictional Eddies remains alone to be con- 
sidered. That resistance is a combination of the direct and indirect 
effects of the adhesion between the skin of the ship and the particles of 
water which glide over it ; which adhesion, together with the stiffness 
of the water, occasions the production of a vast number of small whirls, 
or eddies, in the layer of water immediately adjoining the ship's sur- 



* A sine is one of the measures of an angle. Thus in the cir- 
cle a d c e (fig. 5S) the lines a b and a e are radii of the circle at 
right angles with one another, and c a is the sine of the angle 
c b a, and Dais the sine of the angle d b a The circle is sup- 
posed to be divided into 360 degrees, so that a quadrant, or one- 
fourth of a circle, is 90 degrees. 

In fig. 59 the various trigonometrical quantities relating to 
the angle a are graphically represented. The angle a is half a 
right angle, or 45 degrees, which is the eighth part of the whole 
8ircle of 360 desTees. 



Fig. 58. 




Fig. 59. 



-C OTA N GEM 




RESISTANCE FROM FRICTION AL EDDIES. 44.5 

face. The velocity with which the particles of water whirl in those ed- 
dies, bears some fixed proportion to that with which those particles 
glide over the ship's surface; hence the actual energy of the whirling 
motion impressed on a given mass of water at the expense of the pro- 
pelling power of the ship, being proportional to the square of the velocity 
of the whirling motion, is proportional to the square of the velocity of 
gliding ; in other words, it is proportional to the height due to the ve- 
locity of gliding. The velocity of gliding of the particles of water over 
a given portion of the ship's skin, bears a ratio to the speed of the ship 
depending on her figure, and on the position of the part of her skin in 
question ; and the height due to the velocity of gliding is equal lo the 
height due to the speed of the ship, multiplied by the square of the same 
ratio. Further, the mass of water upon which whirling motion is im- 
pressed by a given part of the ship's skin while she advances through a 
unit of distance, is. proportional to the area of that part of the skin, mul- 
tiplied by the before-mentioned ratio which the velocity of gliding of the 
water past that part of the skin bears to the velocity of the ship. 

Hence the resistance to the motion of the ship, due to the production of 
frictional eddies iy a given portion of her shin, is the product of the fol- 
lowing factors : — 

I. The area of the portion of the ship's skin in question. 

II. The cube of the ratio which the velocity of gliding of the particles 
of water over that area bears to the speed of the ship ; being a quantity 
depending on the figure of the ship and the position of the part of her 
skin under consideration. 

III. The height due to the ship's speed ; that is, 

(speed in feet per second) 2 

(speed in knots) 2 
or, 

22-6 

IV. The heaviness (or weight of a unit of volume) of the water 
(64 lbs. per cubic foot for sea-water). 

V. A factor called the coefficient of friction, depending on the mate- 
rial with which the ship's skin is coated, and its condition as to rough- 
ness or smoothness. 

The sum of the products of the Factors I. and II. for the whole skin 
of the ship has of late been called her Augmented Surface ; and the 
Eddy-resistance of the whole ship may therefore be expressed as the 
product of her Augmented Surface by the Factors III. IV. and V. above 
mentioned.* 



* In algebraical symbols, let d s denote the area of a small portion of the ship's 



Id6 STEAM NAVIGATION. 

The resistance thus determined, being deduced from the work per- 
formed in producing eddies, includes in one quantity both the direct ad- 
hesive action of the water on the ship's skin, and the indirect action, 
through increase of pressure at the bow and diminution of the pressure 
at the stern. 

The existence of this kind of resistance has been recognised from an 
early period. Beaufoy made experiments on models to determine its 
amount ; Mr. Hawksley and Mr. Phipps have included it in a formula for 
the resistance of ships ; and Mr. Bourne pointed out that it must depend 
mainly on the ship's immersed girth. But the earlier researches, both 
experimental and theoretical, throw little light on the subject, and fail 
to give a trustworthy value of the coefficient of friction ; because in 
them it was assumed that the frictionai resistance was proportional to 
the actual immersed surface of the vessel, and the variations of the 
speed of the gliding of the water over different parts of that surface 
were neglected. 

When the Editor of this treatise t (having occasion to compute, in 
1857, the probable resistance at a given speed of a steam-vessel built by 
Mr. J. R. Napier), introduced for the first time the consideration of the 
augmented surface, he adopted, for the coefficient of friction, the con- 
stant part of the expression deduced by Professor Weisbach from exper- 
iments on the flow of water in iron pipes, viz. : 

/= -0036; 

and that value has given results corroborated by practice, for surfaces 
of clean painted iron. For clean copper sheathing, and for very smooth 
pitch, it appears probable that the coefficient of friction is somewhat 
smaller ^ but there are not sufficient experimental data to decide that 
question exactly. Experimental data are also wanting to determine 
the precise increase of the coefficient of friction produced by various 
kinds and degrees of roughness and foulness of the ship's bottom ; but 
it is certain that that increase is sometimes very great. 

The preceding value of the coefficient of friction leads to the follow- 
ing very simple rule for clean painted iron ships: — At ten knots, the eddy - 



skin ; g, the ratio which the velocity of gliding of the water over that portion 
bears to the speed of the ship ; c, the speed of the 6hip ; »/, gravity ; ic, the heavi- 
ness of the vater ; /, the coefficient of friction ; then 

c 2 
Eddy-resistance =/w — fq 3 d «; 

2(7 
fq 2 d s being the Augmented Surface. 

+ The treatise referred to is a 'Treatise on Shipbuilding, , by Mr. Eankine and 
other eminent authorities, in course of publication in 1S65. 



COMPUTATION OF POWER AND SPEED, 447 

resistance is one pound avoirdupois per square foot of augmented surface ; 
and varies, for other speeds, as the square of the speed. 

COMPUTATION OF PHOPELLING POWER AND SPEED. 

General Explanations. — The method of calculation now to be ex- 
plained and illustrated was first practically used in 1857, under the cir- 
cumstances stated. A very condensed account of it, illustrated by a 
table of examples, was read to the British Association in September, 
1861, and printed in various mechanical journals for October of that 
year; and some further explanations appeared in a paper on Waves in 
the 'Philosophical Transactions for 1862.'* 

The method proceeds by deducing the eddy-resistance from an ap- 
proximate value of the augmented surface. It is therefore applicable to 
those vessels only in which eddy-resistance forms the whole of the ap- 
preciable resistance ; but such is the case with all vessels of proportions 
and figures well adapted to their speed, as has been explained in the 
preceding sections ; and as for misshapen and ill-proportioned vessels, 
there does not exist any theory capable of giving their resistance by pre- 
rious computation. 

Computation of Augmented Surface. — To compute the exact aug- 
mented surface of a vessel of any ordinary shape would be a problem 
of impracticable labour and complexity. The method employed, there- 
fore, as an approximation for practical purposes, is to choose in the 
first instance a figure approximating to the actual figure, but of such a 
kind that its augmented surface can be calculated by a simple and easy 
process, and to use that augmented surface instead of the exact aug- 
mented surface of the ship ; care being taken to ascertain by comparison 
with experiments on ships of various sizes and forms whether the ap- 
proximation so obtained is sufficiently accurate. 

The figure chosen for that purpose is the trochoid, or rolling-wave- 
curve, extending between a pair of crests, such as a and b in fig. 60 ; 

Fig. 60. 



for by an easy integration, published in the 'Philosophical Transactions 
for 1862/ it is found that the augmented surface of a trochoidal riband t 

* A prediction of the speed of the ' Great Eastern,' with different amounts of 
engine-power, obtained by this method of calculation, was published in the 
Philosophical Magazine' for April, 1859. 
t This is the species of curve that will be described by a pendulum, the surface 



448 STEAM NAVIGATION. 

of a given length in a straight line, and of a given breadth, is equal tc 
the product of that length and breadth, multiplied by the following co- 
efficient of augmentation ; — 

1 + 4 (sine of greatest obliquity) 2 + (sine of greaiest obliquity) 4 ; the 
greatest obliquity meaning the greatest angle, bed, made by a tangent, 
d e, to the riband at its point of contrary flexure, d, with its straight 
chord, a b. 

In approximating to the augmented surface of a given ship by the 
aid of that of a trochoidal riband, the following values are employed : 

I. For the length, ab, of the riband, the length of the ship on the 
plane of flotation. 

II. For the total breadth of the riband, the mean immersed girth ; 
found by measuring, on the body-plan, the immersed girths of a series 
of cross-sections, and taking their mean by Simpson's Rule, or by 
measuring mechanically with an instrument the sum of a number of 
girths, and dividing by their number. 

III. For the coefficient of augmentation, the mean of the values of that 
coefficient as deduced from the greatest angles of obliquity of the series 
of water-lines of the fore-body, shown on the half-breadth plan. It is 
not necessary to measure the angles themselves, but only their sines. 

The augmented surface is then computed by multiplying together 
those three factors. 

The Computation of the Probable Resistance (in lbs.) at a given speed 
is performed according to the rule already stated, by multiplying the 
augmented surfase by the square of the speed in Tcnots, and dividing by 100 
(for clean painted iron ships). 

The process just described is virtually equivalent "to the following : — 
An ocean wave is conceived (a c b in fig. 60), of a length, a b, equal to 
that of the ship on her water-line ; and having its steepest angle of 
slope, bed, such that the function of that slope, given in Article 162 as 
the coefficient of augmentation, shall be equal to the mean value of the 
same function for all the water-lines of the ship's bow. A solid of a 
breadth equal to the ship's mean immersed girth is then conceived to be 
fitted into the hollow, a cb, and to be moved along with the advance of 
the wave ; and the resistance due to frictional action between that solid 
and the particles of water is taken as the approximate value of the re- 
sistance of the vessel. 

In Computing the Probable Engine Power required at a given Speed, 
allowance must be made for the power wasted through slip, through 
wasteful resistance of the propeller, and through the friction of the en- 

of which was shown by me in my ' Catechism of the Steam-Engine, 1 published in 
1856, to he the measure of the resistance — a conclusion deduced by me from exper- 
iment several years before. 



RULE FOR COMPUTING SPEED. 449 

gine. The proportion borne by that wasted power to the effective or 
net power employed in driving the vessel, of course varies considerably 
.n different ships, propellers, and engines; but in several good examples 
it has been found to differ little from 0'63 ; so that, as a probable value 
of the indicated power required in a well-designed vessel, we may take — 

net power x 1'63. 
Now an indicated horse-power is 550 foot-pounds per second ; ani a 
Knot is 1*688 foot per second; therefore an indicated horse-power is 

550 

= 326 knot-pounds, nearly ; 

or 326 lbs. of gross resistance overcome through one nautical mile in an 
hour. If we estimate, then, the net or useful work done in propelling 
the vessel as equal to the total work of the steam divided by 1'63, we 
shall have 

326 
— — - = 200 knot-pounds 

of net work done in propulsion for each indicated horse-power. Hence 
the following 

Kule. — Multiply tlie Augmented Surface in square feet by the cube of 
the speed in lenots and divide by 20000 ; the quotient will be the probable 
indicated horse-power. 

The divisor in this rule, 20000, expresses the number of square feet of 
augmented surface which can be driven at one Jcnot by one indicated 
horse-power : it may be called the Coefficient of Propulsion. 

It is, of course, to be understood that the exact coefficient of propul- 
sion differs in different vessels, according to the smoothness of the skin, 
the nature of its material, and the efficiency of the engines and propel- 
lers ; being greatest in the most favourable examples. 

In clean iron ships, with no evident fault in shape or dimensions, or 
in the propeller and engine, it has been found on an average to be some- 
what above 20000 ; and the value 20000 may be taken as a probable and 
safe estimate of the coefficient of propulsion in any proposed vessel de- 
signed on good principles. In every instance in which that coefficient 
is materially less than 20000, the shortcoming can be accounted for by 
some fault, such as undue bluntness of the bow or stern. 

In vessels sheathed with copper or coated with smooth pitch, the co- 
efficient of propulsion is unquestionably greater ; but- in what precise 
proportion it is at present difficult to say, owing to the scarcity of ex- 
perimental data. 

Computation of Probable Speed. — When the augmented surface of a 
ship has been determined, her probable speed with a given power is 
computed as follows : — 



450 STEAM NAVIGATION. 

Multiply the indicated Horse-power by the Coefficient of Propulsion (say 
for clean iron ships, 20000) : divide by the Augmented Surface, and extract 
ti cube root of the quotient for the probable speed in knots. 

Example I. — Calculation of Probable Speed of H. M. S. ' Warrior.' 

Displacement on Trial 8997 tons 

tw™^+ ,vPWof«,. {Forward... 25-83 feet. 

Draught of Water "j AT 26 - 75 " 

\Vater-line3. Sine of Obliquitv. Square of Sine. 4th power of Su>«. 

L.W.L -370 -1369 -01874 

2 W.L -315 '0992 -00984 

3 W.L -290 -0841 -00707 

4 W.L -265 -0702 -00492 

5 W.L -235 -0552 .- -00304 

6 W.L -1G5 '0272 -00074 

Keel -000 -0000 00000 

Means -0674 -00583 

1 + (4 x -0674) + -005S3 = 1-275, Coefficient of Augmentation 

Half-girths from Body-plan Simpson's Products. 
Foot. Multipliers. 

210 1 21-0 

27-2 4 108-8 

30-8 2 61-6 

346 4 138-4 

38-8 2 77.6 

41-5 4 166-0 

42-6 2 85-2 

44-0 4 176.0 

44-0 2 88-0 

44-0 4 176.0 

43-3 2 86-6 

421 4 168-4 

40-3 2 80-6 

331 4 152-4 

36-0 2 72-0 

35-0 4 140-0 

32-0 1 32-0 



Divide by : 3)1830-6 Sum". 

Divide by $ number of Intervals 8} 6-012 

Mean Immersed Girth 76-3 

x Length 380 



Product 28994 

x Coefficient of Augmentation 1-275 

Augmented Surface 36979 Square Feel- 
Indicated Horse-power on Trial 5471 

x Coefficient of Propulsion 20000 

Divide "by Aug. Surface 36979)109,420,000 Product 

Cube of Probable Speed 2959 

Probable Speed, computed 14-356 Knots. 

Actual Speed, on Trial 14*354 

Difference '002 



EXAMPLE OF COMPUTATION OP SPEED 451 

Example II. — H. M. S. ' Fairy ' will next be taken as an example, on 
account of the great contrast in size between ber and tbe 'Warrior.' 

Displacement 16S tons. 

Draught of Water 4-83 feet. 

Water-lines. Sins of Obliquity. Square oi Sine. Fourth power of Sink. 
L.W.L "23 -0529 -0028 

2 W.L -22 -04S4 -0023 

3 W.L 21 -0441 -0019 

4 W.L -17 -02S9 -0008 

Keel 



Means -0304 -0015 

1 + (4 x -0304) + -0015 = 1-123, Coefficient of Augmentation. 

Length on Water-line 144 Peet. 

x Mean Immersed Girth (measured mechanically with 

an Instrument 19 

x Coefficient Augmentation 1-123 



Augmented Surface 3072 Square Peet. 

Indicated Horse-power, on Trial 3&4 

x Coefficient of Propulsion 20000 



-*- Augmented Surface 3072)7,280,000 Product. 



Cube of Probable Speed 2370 

Probable Speed, computed 13-333 Knots. 

Actual Speed, on Trial 13*324 



Differenco -009 

Tbe ' Fairy ' occurs in tbe table of examples given in tbe paper of 
18G1, already referred to : in tbe present paper tbe measurements bave 
been revised and improved in precision, especially as regards tbe co- 
efficient of augmentation. The difference in the result is but small. 

Example III. — H. M. S. ' Yictoria and Albert ' — a wooden vessel, 
sheathed with copper, will now be employed, not to illustrate tbe com- 
putation of probable power at a given speed, or of probable speed at a 
given power ; but to compute a value of the coefficient of propulsion 
for a copper-sheathed vessel. 



Displacement on Trial Trip 1980 tons. 

' >8 feet 
•0 " 



Draught of Water " j i£^- " [ it 



Water-lines. Sine of Obliquity. Square of Sine. 4th power of Sine. 

L.W.L -19 -0361 -0013 

2 W.L -1S5 -0342 -0012 

8 W.L -17 -02S9 -0008 

4 W.L -14 -0196 -0004 

Keel 



Means '0252 -0003 

1 + (4 x -0252) + -0008 ~ 1-102, Coefficient of Augmentation. 



4:52 STEAM NAVIGATION. 

Length on Water-line 300 Feet. 

xMean Immersed Girth (measured mechanically with 

an Instrument) 40 

x Coefficient of Augmentation 1*102 

Augmented Surface 13224 Square Feet. 

x cube of Speed in Knots IT 3 = 4913 

-j- Indicated Horse-power on Trial ^30)64,969,512 Product. 

Coefficient of Propulsion 21,802 

Had the probable speed been computed with the coefficient of propulsion 
20000, the result would have been 16-53 knots, instead of 17. 

Proportions of Length to Breadth. — Principles which have been al- 
ready explained fix the least absolute length suitable for a vessel which is 
to be driven at a given speed. But after that least length has been 
fixed, a question may arise as to whether that least length, or a greater 
length, is the most economical of power. That question is answered by 
finding the proportion of length to breadth, which gives the least aug- 
mented surface with the required displacement. 

That proportion can be found in an approximate way only; because 
of the approximate nature of the process by which the augmented sur- 
face itself is found. The following are some of the results obtained in 
certain cases : — 

I. When the proportion of breadth to draught of water, and the 
figure of cross-section, are fixed, so that the mean girth bears a fixed 
proportion to the breadth, it appears that the proportion of length to 
breadth which gives the least augmented surface for a given displace- 
ment, is about 7 to 1. 

II. "When the absolute draught of water is fixed, the proportion of 
length to breadth which gives the least augmented surface for a given 
displacement depends on the proportion borne by the draught of water 
to a mean proportional between the length and breadth, and on tho 
figures of the cross-sections. The following are some examples for flat- 
bottomed vessels : — 

( ^Length x Breadth) „ H 

H 2. from 4 to 5 ; 7 to 10 : 12 to 16 : 1 1 to 23 : 

Draught ' ' ' 

^\ *i 8; 9; 10. 

Breadth ' ' ' 

III. By cutting a vessel in two amidships, and inserting a straight 
middle body, the proportion borne by her resistance to her displace* 
ment is always diminished ; because the midship section has a less 
mean girth in proportion to its area than any other cross-section of the 
ship ; and therefore the new middle body adds propo.'tionallj less t<? 
the augmented surface than it does to the displacement. 

IV. It does not follow, however, that a straight middle between 
tapering ends is tie most economical form: for by adopting continuous 



SUMMARY OF MAIN DOCTRINES- . 453 

curves from bow to stern for the water-lines, instead of the lines com- 
pounded of curved ends and a straight middle, the same length, the 
same displacement, and almost exactly the same mean girth may be 
preserved, and the obliquity of the water-lines at the entrance dimin- 
ished. 

GENERAL CONCLUSIONS. 

The principal conclusions to be drawn from the foregoing ex- 
position are the following : — 

1st. That the bulk of a ship should be equal to half the bulk of 
the circumscribing parallelepiped, supposing the areas of all the 
cross-sections have been translated into the form of a rectangle. 

2d. That the sectional area of each successive frame should 
vary as the square of the distance from the stem or stern, until 
the points midway between the midship frame and the stem or 
stern have been reached, and that the areas at all the frames 
should vary in the manner already pointed out. 

3d. That it is better to place the midship frame before the 
centre of the ship, in order that any wave raised at the stern 
may be sufficiently far forward to assist the propulsion. 

4th. That the horizontal water-lines should be pendulum or 
trochoidal curves, or such equivalent curve as will enable the 
progressive displacement to follow the prescribed law, and that 
the transverse section should be formed with similar curves made 
as nearly as possible coincident with a semicircle. 

5th. That nearly the whole of the resistance in a well-formed 
vessel is made up of friction, and that the friction per square 
foot of surface is less at the stern than at the bow, but that the 
law of variation is not known. Also, that at a certain point of 
the length the water adhering to the ship will attain its maxi- 
mum velocity, and thereafter every foot of the length will have 
the same resistance. 

6th. That the friction is diminished by making the bottom 
fair and smooth, and by coating it with a suitable lubricant, and 
that a portion of the power expended in friction may be recov- 
ered by making the stern part of the vessel to overhang near (ho 
water-line, so as to be propelled by the upward motion of the 
current which the friction generates, and also by placing the 
rropeiler in the stern or quarters instead of at the sides 



454 



STEAM NAVIGATION. 



7tli. That both, by Boulton and "Watt's method and by Mi*. 
Rankine's method the speed of a steamer may be accurately pre 
dieted. Boulton and "Watt, by whom the screw engines of the 
4 Great Eastern ' were made, predicted that the speed of the ves- 
sel would be 16-57 statute miles with 10,000 actual horse-power. 
"Not more than 8,000 horse-power were actually generated, in 
consequence of a deficiency of steam. But on trial the speed 
was found to be as nearly as possible what it ought to be accord- 
ing to their rule with this proportion of power. The coefficient 
they employed for statute miles in making this computation was 
900, which is also the coefficient they habitually use in the case 
of fast river boats of considerable size and good form. 

8th. That any expedient for diminishing the resistance of 
well-formed vessels to be of material efficacy must have for its 
object the diminution of the friction of the bottom, either by re- 
ducing the adhesion of the particles of water to the ship, or to 
one another, or both ; and also by mating the adhering surface 
as small as possible. 

EXAMPLES OF LINES OF APPROVED STEAMERS. 

In fig. 61 we have the body-plan of H. M~. screw yacht 'Fai- 
ry,' 144 feet 8 inches long between perpendiculars ; and the hor- 

Fia:. 61. 




BODY PLAN OF H.M.S. * FAIRY. 



izontal water-lines can easily be constructed from the body-plan, 
by dividing the length by the number of vertical lines or frames, 
*nd by setting off at each division the given breadth of each wa- 



STEAMERS * RATTLER ' AND 'BREMEN.' 455 

ter line at that part. The c Fairy ' is 312 tons, 21 feet 1^ inch 
extreme breadth, and has 74*4 square feet of immersed section at 
5 feet draught. She is propelled by two oscillating geared en- 
gines of 42 inches diameter of cylinder and 3 feet stroke, and has 
attained a speed of 13*3 knots per hour, exerting 363*8 indicated 
horse-power. 

The 'Battler' is 176 feet 6 inches long between perpendicu- 
lars, 32 feet 8|- inches extreme breadth, 888 tons burden, 804 
tons displacement at 11 feet 5-|inches mean draught, 281*8 square 

Fig. 62. 




BODY PLAX OF H. M. S. ' BATTLER. 



feet of immersed section, and is propelled by geared engines of 
200 nominal horse-power. With 428 indicated horse-power she 
attained a speed of lOJmots — a high result, imputable partly to 
her good form for such speed, and partly to the smoothness of 
the copper sheathing— the 'Battler' being a wooden vessel. 

In the steamer ' Bremen,' which has given a very favourable 
result in working, the length of keel and fore-rake is 318 feet ; 
the breadth of beam 40 feet; depth of hold 2G feet; tonnage, 
builders measurement, 2,500 tons ; power, two direct-acting in- 
verted cylinder engines, with cylinders of 90 inches diameter 
and 3-£- feet stroke. With a draught of 18-J- feet the displace- 



456 



STEAM NAVIGATION. 



ment was 3,440 tons, the area of immersed section 606 square 
feet, and with the engines working to 1,624 horse-power the 
speed attained was 13'15 knot;. 

Fig. 64 is the body plan of the Ctniard steamer 'PeisiaJ 
The vertical sections are 17-i- feet from one another, and the 
breadth of the vessel is 45 feet. The engines are side lever : 
cylinders 100 inches diameter and 10 feet stroke, making 18 
strokes per minute. The laily consumption of coals in eight 
boilers containing 40 furnaces is 130 tons, and the pressure of 

Kg. 63. 




BODY PLA>" OF SCEETV STEAUXE 'BKEMEK". 



the steam is 25 lbs. per square inch. The performance of the 
'Persia' has been very satisfactory, except that she was not 
strong enough in the deck and had to be strengthened there, and 
she has a great deal too much iron in the shape of frames, which 
conduce to weakness rather than to strength. The paddle- 
wheels are 40 feet in diameter, and the "floats are 10 feet long 
and 3 feet wide. 

In fig. 65 the body plan of the iron-plated steamer k Warrior ' 
is uiven. and 66 is a transverse section of the same vessel, show- 
m<y the guns. The 4 Warrior ' is an iron-clad steamer of 
tons, 380 feet long. 53 feet broad, and 1,250 horse-power; and 
with the exertion of 5,460 actual horse-power, and at 2 G feet 
draught, and with an area of immersed section 1,219 square feet, 



STEAMERS ' PERSIA ' AND ' WARRIOR.' 



457 



she realized a speed of 14-3 knots. The utility of such vessels as 
the ' "Warrior ' does not promise to be considerable, and in fact 
the whole idea of constructing ships that would be impenetrable 
to shot turns out to be a complete delusion, as was plainly per- 
ceived by a number of competent observers would necessarily be 




BODY PLAN OP PADDLE STEAMER * PERSIA. 



the case before the expensive demonstrations were resorted to 
which the Admiralty has thought fit to institute. If there had 
been any natural law which restricted the penetrating power of 
ordnance to the narrow limits hitherto existing, there would 
have been some reason in the conclusion that by making the iron 
sides of ships very thick the shot would have been prevented 
20 



458 



STEAM NAVIGATION. 



from penetrating them. But two facts were quite well known . 
first, that a steel punch may be made to pierce an iron plate how- 
ever thick, if the diameter of the punch be equal to the thick- 
ness of the iron ; and, secondly, that by increasing the dimen- 
sions of the gun an amount of projectile force could be obtained 
that would suffice for the punching through of an)- thickness 
whatever. The amount of this force, and of the dimensions of 

Fia:. 65. 




BODY PLAN OF H. M. S. ' WARRIOR.' 



gun requisite to produce it, are of perfectly simple computation. 
The punching pressure is about the same as that required to tear 
asunder a bar of iron of the same sectional area as the surface 
cut by the punch, which is about 60,000 lbs. per square inch: 
and this pressure must act through such a distance as will suffice 
to overcome the continuity of the metal. : The distance through 
which iron stretches before it breaks is quite well known, and 
this distance, multiplied by the separating pressure per square 



FUTILITY OF EXISTING ARMOUR. 



459 



Jnch, gives a measure of the power required. The velocity of 
cannon balls is also known, and, by the law of falling bodies, the 
height from which a body must descend by gravity to acquire 
that velocity can easily be determined ; and the weight of the 
ball in lbs., multiplied by this height in feet, must always bo 
greater than the punching pressure in lbs. multiplied by the dis- 
tance in parts of a foot through which iron stretches before it 
breaks, rise the ball will not penetrate. "We have by no means 

Fig. 66. 




TRANSVERSA SECTION OP H. M. S. ' WARRIOR.' 

reached the limit of the power of projectiles, nor is the explora- 
tion of those limit. 1 ? yet begun. Piston guns may be made in 
which the projectile would consist of a cigar-like body, or thun- 
derbolt, with spiral fins supporting a wooden piston or wad, 
which would transmit to a projectile of small diameter the pow- 
er generated in a cylinder of large diameter. A gun is virtually 
a cylinder, and the ball is the piston; and the power gifen to 
the ball will be represented by the pressure exerted by the ex- 



460 STEAM NAVIGATION. 

ploding powder multiplied by the capacity of the gun. As, how- 
ever, there are practical limits to the length of a gun, it may bo 
advisable to increase the diameter, in order to get the requisite 
power. But this must be done without increasing the diameter 
of the ball, which would encounter greater resistance if made too 
large ; and piston guns are the obvious resource in such a case — 
the piston being so contrived that it would be left behind by the 
ball so soon as it had left the mouth of the gun, and had acquir- 
ed all the power which a piston could communicate. The pro- 
jectile itself should have a sustaining power as well as a pro- 
jectile one, to which end it should contain a certain quantity of 
rocket composition that would burn during the flight of the ball ; 
and as the velocity of the ball would be high, the rocket gas 
would operate with little slip, and with much greater efficiency, 
therefore, than in rockets. The spiral feathers would cause the 
projectile to revolve in its flight, in the same manner in which a 
patent log is turned by the water ; and any need for rifling the 
gun would thus be obviated, as the air would act the part of the 
rifle grooves. By these means far greater ranges and far greater 
accuracy of aim may be obtained than is at present possible, and 
it needs no great perspicacity to see that the success of maritime 
warfare will henceforth depend on the speed of the vessels em- 
ployed, and the range, force, and accuracy of the projectiles. A 
small and very swift steamer with projectiles of the kind I have 
described would be able to destroy at her leisure a vessel like the 
' Warrior,' while herself keeping out of range of the best existing 
guns which the assailed vessel could bring to bear against her 
opponent. "With great accuracy of aim, and by choosing a posi- 
tion where the wind would have little disturbing influence, a 
large vessel could be struck at a distance at present deemed chi- 
merical, and a few of such vessels as I have described, without 
any armour at all, would speedily disable any vessel which was 
not provided with the same species of projectile. Even if the 
large vessels, however, were to be armed with projectiles of 
equal range and power, the advantage would still be with the 
small vessels, as they would be more difficult to hit ; and by 
taking up an external position and firing their guns in converg- 



FEATURES OF AMERICAN MONITORS. 461 

mg lines, of which the assailed object would be the focus, a great 
advantage would be given in the attack. 

The vessels called Monitors, recently constructed in America, 
and which, I believe, owe their most valuable features to the 
talents of Ericcson, the eminent Swedish engineer — whose ser- 
vices were lost to this country through the incapacity of the Ad- 
miralty at the time of the introduction of the screw-propeller — 
are a very judicious embodiment of the leading principles of iron- 
clad vessels so as to secure the greatest possible efficiency. The 
constructors of those vessels saw that the thickness of the side? 
must be very much greater than it is in our iron-clads, to pre- 
vent heavy shot from going through them ; and this thickness is 
reconciled with the usual buoyancy by making the sides of the 
vessel very low, so that only a small area has to be protected. 
Yery powerful guns are employed in these vessels; and as it 
would be difficult to manoeuvre such guns by hand, a steam-en- 
gine is introduced for this purpose, which gives great facility in 
the handling. To protect the guns and gunners from hostile 
shot, they are placed in towers of iron, the metal of which is 15 
inches thick, and these towers are turned like a swing-bridge to 
enable the gun to be pointed ; but the mechanism is so contriv- 
ed, that the hand of a child acting on the engine will suffice to 
move the tower. Admiral Porter states that a Monitor of this 
construction would be able to cross the Atlantic, and attack and 
sink our iron-clads at her leisure, without being herself liable to 
injury ; and I think he is right in his conclusion, though it was 
a most indelicate thing for him to have indicated such an occu- 
pation for this class of vessels. But persons who infer the help- 
lessness of this country to resist such attacks, from the imbecili- 
ty of the Admiralty, will find themselves mistaken ; and there 
are obviously two ways in which such Monitors could be de- 
stroyed. Those vessels, though immensely strong above the 
water, are weak below, being there without armour, as they are 
protected from shot by the water. But a vessel like the ' War- 
rior,' if armed in a line with the keel — or a little above it — with 
a great steel blade or horn 40 or 50 feet long, would by running 
against a Monitor, break into the bottom and sink her. Such a 



462 STEAM NAVIGATION. 

conflict would "be like a sword-fish attacking a whale ; and the 
horn or blade would in no way affect the steering of the vessel, 
as it would only virtually make her so much longer. Another 
way in which Monitors could be destroyed, is by running over 
them. As they are not many feet out of the water, to submerge 
them for a few feet more, by placing a corresponding weight 
upon their deck, would sink them altogether ; and if we suppose 
a vessel with a very raking stem, and so trimmed by the stern as 
to bring the forefoot out of the water, to be run against a Mom- 
tor, it will be obvious, if the vessel be a large and heavy one, and 
the speed of propulsion be high, that she would run up on the 
deck of the Monitor, and sink her at once. The weight and 
speed of vessel that would work this catastrojihe in the case of 
any given Monitor, is matter of simple caiculation; and it is 
quite an error, therefore, to imagine that any Monitor yet con- 
structed might not be promptly disposed of. Certainly they 
might be made tight, like diving-bells, so that even if sunk and 
ridden over, they would come up again. But this would be a 
difficult tiling to do ; and even if it were done, the next step 
would be, that the attacking vessel would not go over, but would 
stop upon them. ~Ko doubt the Monitor might as easily run 
into the attacking vessel as the attacking vessel into her, pro- 
vided the Monitor had equal speed. But the construction of 
Monitors is not favourable for speed ; and if speed is to settle 
the question, there is no need for iron plating. The fact is, such 
infallible recipes for victory as Monitors are supposed to consti- 
tute, almost always break down. I believe that such vessels 
may be made sea-worthy ; they may be made impenetrable to 
any guns at present in our navy, and the grins they mount may 
be able to riddle our iron-clads like so many ships of card-board. 
All that I grant. But guns can be made to go through the tow- 
ers and sides of Monitors, though twice as thick as they are all 
the existing Monitors can easily be outstripped in speed; and 
vessels with steel horns may rip up their bottoms, and ves- 
sels built with greatly slanted stems may be made to run over 
and sink them. It is true there are the guns of the Monitor to 
be encountered by the attacking vessel. But if that vessel has 



FEATURES OF AMERICAN MONITORS. 



4G3 



"V . 




464 



STEAM NAVIGATION. 



several decks, and if the deck over the main hold be made into 
a water-tank, with water-tight trunks communicating between 
the hold and the decks above, a shot between wind and water 
would not let water in, as the space is filled with water already; 
and the attacking vessel, therefore, could not be sunk by any fire 
the Monitor could bring against her, unless it could be made to 
pierce through the sea so as to enter the lower hold by whicr 
the flotation is given. "With the low elevation of the Monitoi 
turrets, however, this does not appear to be a probable contin- 
gency. Small rocket-vessels, propelled at a high speed by rock- 
et gas issuing at the stern beneath the water, will probably be 
used in actual warfare for many purposes ; and the same resource 
may be employed temporarily to increase the speed of large 
steamers. If, for example, the iron-clads of the ' "Warrior ' type 
had a tube opening beneath the water at each quarter, out of 
which rocket flame and gas were made to issue, the speed of the 
vessel, while the emission lasted, would be increased ; and this 
temporary acceleration might suffice to give her a decisive supe- 
riority oyer an opponent. 





INDEX. 



ABS 

ABSOLUTE zero, 137 
Addition, nature of, 10 ; addition 
table, 11 ; method of performing, 11 ; 
examples of, 12 

— of fractions, 34 

— indicated by + or plus, 10 
Air, composition of, 174 

— dilatation of, by heat, 145, 147 

— height of column of, to produce atmos- 
pheric pressure, 100 ; relative density 
of, 101 

— into a vacuum, velocity of, 101 

— in water lowers the boiling point, 168 

— pump and condenser, proportions of, 
215, 222 

indicator diagrams taken from, 344, 

345, 351, 357 

studs in side levers, 269 

of marine engines, proper propor- 
tions of, 2S0 

rod of marine engines, proper pro- 

tions of, 2S0 ; table of proportions of, 
299 

side rods of marine engines, 284 

■ rod of land engines, 232 

■ crosshead, proper dimensions of, 

282 

■ bucket, cutter of, 281 

Algebra, wherein it differs from arithme- 
tic, 10 

Allen's engine, diagrams from, 357 

American monitors, 461 

Annular valves, how to compute the 
pressure on, 212 

Appold's centrifugal pump, 386 

Archimedes screw, 386 

Arithmetic of the steam-engine, 1 

— defined, 5 

•- wherein it diifers from algebra, 10 



BOI 

Armour of ships of war penetrable, 458; 
measure of its resistance to shot, 459 

Atmospheric pressure, height of column 
of air required to produce, 101 

Attraction of gravity, 93 

Augmented surface of a vessel a measure 
of resistance, 443 

Auxiliary propulsion of common steam- 
ers by rockets, 464 



BACK links, 232 
Barley mill, 387 
Barlow's experiments on the strength of 

woods, 127 
' Barossa,' diagram from, 350 
Battering ram, momentum of, 105 
Beam of land engines, 233 
Beams, how to determine strength of, S7 

— cast-iron, Hodgkinson's rule for 
strength of, 133 

Bean mill, 388 

Bearings, friction of, how to limit, 121; 
variations of velocity and pressure, 

Blast orifice in locomotives, area of, 313 

— pipe in locomotives, 331 

Block and tackle, weights movable by, 

82 
Bochet's experiments on friction, 119 
Bodies, falling, laws of, 93, 97 

— revolving, centrifugal force of, 109 ; 
bursting velocity, 110 

Body-plan of steamer 'Fairy,' 454; of 
' Battler,' 455 ; ' Bremen,' 456 ; ' Persia,' 
457; ' Warrior,' 45S 

Boilers, circulation of water in, very inv 
portant, 173 

— proportions of, 303 



466 



LSDEX. 



BOI 

Boilers, power measurable by evapora- 
tion, 309 

— wason. proportions of. 310 : Sue, 311 

— haystack, by D. Xapier, 316; by Earl 
of Dundonald, 316 

— strength of, 320, 322 

— stays, 321 

— cylindrical, proper diameter for given 
pressure of steam and thickness of 
plate, 324 ; safe pressure in, 325 

— bursting and safe working pressures 
of, 326 

— of modern construction, heating sur- 
face of. 375 

— uptake, sectional area of, 376 

— and surface condensers, relative sur- 
face areas of. 380, 382 

— fed by Giffard's injector, 353 

— marine, bulk of, 313 

— locomotive, example of. 329 
Boiling point of water raised by molec- 
ular "attraction, 16S 

lowered by presence of extraneous 

substances, 163 

Boulton and "Watt's rule for the fly- 
wheel, 22S 

system in drawing office, 259 

Bourne's "duty meter. 373 

Boutigny, his" experiments on spheroidal 
condition of liquids, 169 

Breadth, maximum, of ships, best posi- 
tion of. 417 

' Bremen,' steamer. 456 

Brule, membrane pump by. 355 

Bucket of air-pump, cutter through, 
251 

Bulk of marine tubular boilers, 313 

Bursting and safe working pressures in 
boilers, 326 

— velocity of fly-wheels, 110 



CAIED and Co., dimensions of side le- 
ver marine engines by. 257 

engines of ' Hansa ' by. 314 

Canals, velocity of water in. ] 65, 433 
Cannon ball, momentum of, 105 
Carbonic acid, specific gravity of, 175 

— oxide produced in bad furnaces, 176 
Carl-Metz, pumps by, 355 

Cast-iron, limit of load on in machinery, 

BS 

beams, proportions of. S3 

columns, strength of, 131 ; beams. 

133 
Centigrade thermometer, 133 
Centres of gyration and percussion, 112 
oscillation, 115 

— in side lever, 269 

Centrifugal force. 107: how to determine 

the, 10s; bursting velocity. 110 
Centrifugal pendulum or Governer, 117 

— pumps, 356 



COS 

Characteristic in logarithms, nature of, £5 
Cheapest source of rower. 151 
Chelsea Water Works, engines of. S6S 
Chimneys, exhaustion produced by. 304, 
Boulton and Watt's rule for proportion 
of, in land boilers. 305, 313 ; in marine 
boilers, 305, 314 ; Peclefs rule for pro- 
portions of, 306 

— proper height of. 305 

— sectional area of, required to evaporate 
a cubic foot per hour. 315 

Circular and square inches, 8 
Circulation of water in bodies very im- 
portant. 173 

heat, 171 

Circular saw. 359 

— loom, 393 

Circumscribing parallelopiped. 4(>G 
'Clyde' steamer, dimensions of. 257 
Coals, heating powers of different, 177 

— consumed per square foot of fire bars 
to evaporate a cubic foot per hour, 314, 
316 

indicated horse power per hour, 

at Chelsea Water Works, 368 
Coefficient multiplier, 77 

— of friction, 119 

Coefficients of various steamers, 77 

— of dilatation of gases, 144 
Cohesion of water, 163 

Coke burned in locomotives, 331 

Collapsing pressure of flues, 327 

Colours, how produced. 94 

Columns, laws of strength of, 123, 131 

Cold water pump, to find the proper ca- 
pacity of. 227 

Combustion, nature of, 174: air required 
for, 174 ; total heat of. 176 ; rates of, 
179 

Combustibles, evaporative powers of, 
175 

Common divisor defined, S3 

— denominator, how to reduce fractions 
to, 35 

Compound quantities. 57 
Compressibility of gases. 143 
Conical measure, 9 
Condensation of steam by cold surfaces, 

173 ; secret of refrigerative efficiency, 

173 
Condenser and air-pump, proportions of, 

215 

— and boiler surfaces, proportionate areas 
of, 173 

Condensers, proper construction of. 315 

— surface, cause internal corrosion in 
boilers, 3S1; proportions of, in recent 
cases. 352 

Conduction of heat, 172 
Conducting powers of metals. 172 
Conservation of energy. 7? 

— force. 7S 

Connecting rod of land engines, 238 



INDEX. 



467 



CON 

Connecting: rod of marine engines of 
wrought-iron, 263 

Consumption of fuel at Chelsea Water 
Works, 368 

.— of coal in steamer ' Han so, '315 

Cooling surface of condensers, 315 

Corrosion of boilers internally with sur- 
face condensers, 381 

Cotton spinning mill, 391 

Counter, 372 

Cranes, weights lifted hy, 82 

Crank, btrain from infinite, 90 

Crank, shafts of cast-iron, Mr. Watt's rule 
for, 239 

— large eye of, when cast-iron, 240 

— when of cast-iron, 242 

— table of proportions of, 296, 29T, 298, 
300 

— pin, when of cast-iron, 246 
journal, 271 

of marine engines, tabic of propor- 
tions of, 300 

Cranks for marine engines of wrought- 
iron, 271, 278 

Crosshead of marine engines, proper pro- 
portions of, 255 

depth of, rule for, 256 

— eye of, 257 

— of air pump, proper dimensions of, 
2G2 

Crosstail, proper proportions of, 267 
Cress section of ships, best form of, 409 
Crucible, red hot, ice made in, 170 
Cube roots, nature of, 48 
of fractions, 48 

— root, method of extracting, 50 
Cubes and cube roots, 48 
Cubic measure explained, 76 
Cushioning, diagrams showing, 355, 356 
Cutting off the steam, advantage of, 182 
Cutter through piston, 263 

air-pump bucket, 2S1 

— of connecting rod, 265 

Cutters and gibs. See Gibs and Cut- 
tecs. 

Curves, mode of representing dimensions 
by, 2S8 ■ 

Cylindrical measure, 9 



DARCTS experiments on the friction 
of various surfaces in water, 439 
Decimal system of numeration, 2 
Decimal fractions, nature of, 
Denominator of fractions defined, 6 
— common, how to reduce fractions to, 

85 
Density of water, maximum, 139 

steam of atmospheric pressure, 102 

Densities of gases, 166 
Diagrams, indicator, how to read, 337 ; 
how to take, 370; various examples of, 
338; from air-pump, 344, 351, 357 ; from 



EVA 

hot well, 360, 362 ; from water pump, 
361 ; double cylinder, 869 — showing 
momentum of indicator piston, 347 

Diameter of cylindrical boiler proper for 
given pressure and thickness of plate, 
325 

'Dictator' steam ram or monitor, con- 
structed in America by Ericsson, 463 

Differential motions for raising weights, 
85 

— gearing, 86 

Dilatation, 140 ; force of, 143 ; of gases, 

144,145 
Dimensions of engines laid down to 

curves, 288 
side lever marine engines, 254, 

301 
marine engines by Caird, 287 ; by 

Maudslay, 290 ; by Seward, 292 

locomotive engines, 301 

Disc, revolving power resident in, 111 
Divisor defined, 24 

— common, defined, 33 
Dividend defined, 24 

Division, nature of, 24; examples of, 26; 
explanation of, 29 

— of fractions, 38 

Donny, his experiments upon ebullition, 
168 

Double cylinder engines, diagrams from, 
362, 365, 869 

Drawing of one engine suitable for 
another by altering scale, 287 ; conven- 
ient sizes for drawings, 2S9 

Duke of Sutherland's yacht, diagrams 
from, 257, 260 

Dundonald, Earl of, boilers by, 316 

Duty of engines at Lambeth Water- 
works, 368 

Duty meter, 373 

Dynamical unit, 79 

Dynamometer, 372 



EBULLITION, 168 
Elastic force r>f steam at different 

temperatures, 159 
Elbow-jointed lever, 89 
Elliot, Brothers, indicator by, 335 
Energy, conservation of, 78 
Engines, if perfect, power producible by, 

181 
Equations, nature of, 74 
Equation for determining the speed of 

steamers, 76 
Equivalent, mechanical, of heat, 91 
Ericsson the designer of the American 

monitors, 461 
Evaporation, latent heat of, 152 

— in locomotives, 381 

Evaporative powers of combustibles, 
175 

— power of coal, 312 



463 



INDEX, 



EXH 

Exhaustion of chimneys, 304 
Expansion of air by heat, 146 

— of gases, 166 

— by link, diagrams showing, 355, 356 

— of steam, 1S2 ; measure of benefit from, 
1S3 ; mean pressure of expanding steam, 
185 

— producible by a given proportion of 
lap to stroke of valve, 187, 1S8 

Expansion producible by throttling the 

steam, 198 
Exponents, fractional, 51 
Eyes of cranks of wrought-iron, 271, 272 
Eye of air-pump crosshead, 2S3 



FACTOES defined, 29 
Fahrenheit's thermometer, 133 
'Fairy' steamer, lines of, 454 
Falling bodies, laws of, 90, 97 
Fans, power required to drive, 391 
Feed pipe, rule for proportioning, 221 

— pump, to find the proper capacity of, 
224 

Feeding boilers by Giffard's injector, 

3S4 
Film of water moving with a ship, 428 
Fire bars of locomotives, 314 
Fishes, shape of, translatable into that of 

ships, 407 
Flaud, pumps by, 3S5 
Flax mills, 395 

Floating bridge, diagrams from, 355, 356 
Flour mill, 387 
Flues, proper sectional area of, 310, 311 

— boilers, proper proportions of, 311 
Flues, sectional area of, required to 

evaporate a cubic foot of water per 
hour, 314, 

— collapsing pressure of, 827 
Fluids, motion of, 100 

Fly-wheels, momentum of, 106 ; burst- 
ing velocity of, 110 

— should have power equal to six half 
strokes, 216 

Fl v-w heel, Boul ton and "Watt's rule for 
the, 228 

— shaft, 239, 240 

Force, conservation of, 78 

— centrifugal, 107 ; how to measure, 10S ; 
bursting velocity, 110 

— -jf dictation, 143 

— olastic, of steam at different temper- 
atures, by M. Eegnault, 159 

Form of least resistance in ships, 402 
Formula for determining the speed of 

steamers, 76 
Foot valve, passages to find the proper 

area of, 223 
Fractions, nature of, 5; vulgar, 5; deci 

iral, 6 

— multiplication by, 9 

~ nature ancl properties of, 31 ; how to 



GEA 

reduce a fraction to its lowest terms, 
33 

— addition and subtraction of, 34 

— how to reduce a common denominator, 
35 

— multiplication and division of, 38 

— squares and square roots of, 45 

— cubes and cube roots of, 48 

— resolvable into infinite series, 66 
Fractional exponents, 52 

Frame, midship, of ships, best position 
of, 417 

Franklin Institute, experiments on 
steam by, 157 

French Academy, experiments on steam 
by, 157 

Friction, 118; coefficient of, 119; experi- 
ments on, by Morin and Bochet, 118 

— of crank pins, 120 

bearings varies with the pressure 

121 ; relations of pressure and velocity 

122 
Friction of flowing water, 199 

engines, 367 

water in pipes does not vary with 

the pressure, 429 
bodies moving in water varies with 

nature of surface, 439 
the bottom the main source of re 

sistance in ships, 423 

bottom of steamer, ' Leinster, 1 438 

Fuel, different kinds of, heating power, 

175 

— consumed per indicated horse powei 
per hour at Chelsea Water-works, 368 

Fulling mills, 394 
Furnaces, temperatures of, 17S 
rates of combustion, 179 

— importance of high temperature of, 377 



GAS into a vacuum, velocity of, 102 
Gases, dilatation and compression of, 
143, 144, 145 
— , and vapours, difference between, 153 

— liquefied by cold and pressure, 153 

— specific heats of, 164, 165; densities, 
volumes, and rates of expansions of, 
166 

Gearing, differential, 86 

Gearing, proportions proper for, 231 

Gibs and cutters of crosshead, 258 

side rods. 260 

through crosstail, 266 

air-pump crosshead, 280 

air-pump side rods, 2S6 

Giffard's injector, 3S3 

Glass Avorks, 397 

Gf *~ eruor for steam engines, 117 

— to determine the right proportions of 
the, 231 

Grate coal burned on each square foot in 
different boilers. 311 



INDEX. 



469 



GRA 

Grate surface to evaporate a cubic foot 
per hour, 314 

— bars per nominal horse-power in 
steamers, 815 

Gravity, nature of, 93 

Gudgeons in side lever, 269 

Guns, piston, 460 

Gyration, centre of, 112 ; to fiud the posi- 
tion of, 113 

Gyroscope, phenomena of the, 93 

Gwynn's centrifugal pump, 386 



HANSA' steamer, proportions of 
machinery of, 314 
Haystack boiler, 316 
Heat, motive power of, 90 

— mechanical equivalent of, 91, 167 
power producible by, 131 

— sensible, denned, 135 

— latent, defined, 135 

— specific, defined, 135 

— dilatation by, 140 

— specific, 162 

— unit of, 162 

— effect of, in accelerating the velocity 
of rivers, 425 

Heating surface of boiler per square foot 
of fire grate, 314 

to evaporate a cubic foot of 

water per hour, 314 

and cooling surface of con- 
denser, 315 

in modern boilers, 375 

Height from which bodies have fallen 
determinable from their velocity, 98 

Height from which bodies have fallen 
determinable from their time of fall- 
ing, 9S 

— of chimney proper for different boilers, 
305 

Hodgkinson, strength of woods accord- 
ing to, 128 ; law of strength of pillars 
by, 128, 131 ; of cast-iron beams, 133 

Horse-power, nominal, definition of, 79 

actual, definition of, 79 

Hot well, indicator diagrams from, 359, 
860 

Hydraulic press, pressure producible by, 
81, S4 

— head of water different from hydro- 
static head, 426 

— mean depth of a ship, 431 
Hydrostatic resistance of vessels increas- 
es with speed and with breadth, 422 

— head of water different from hydraulic 
head, 426 



TOE, weight of at 82°, 140 
JL — made in a red-hot crucible, 170 
Improvements required in boilers and 
condensers, 379 



LES 

Inches, square and circular, spherical. 

cylindrical, and conical, 9 
Incommensurables, nature of, 46 
Indian system of numeration, 3 
Indicator, construction of the, 333; 

Eichards', 334; method of applying the, 

335, 870 

— diagrams, how to read 335 ; how to 
take, 370; various examples of, 338; 
from air-pump, 344, 351, 357 ; from 
hot well, 359 ; from water pump, 361 ; 
from double cylinder engine, 365, 
369 

Indicator diagrams, showing momentum 

of indicator piston, 347 
Inertia defined, 105 
Infinite series, how to resolve fractions 

into, 66 

— strains from crank and elbow-jointed 
lever, 89 

Injection pipe, to find the proper area of, 

222 
Injector, Giffard's, 551 
Invisible light, 96 
Iron, steel, and other metals, strength 

of, 125 

— fusible at low temperatures, 151 

— works, 397 

Iron-clad steamers penetrable by shot, 

457 
Irrational numbers defined, 46 
' Island Queen, 1 indicator diagram from, 

339 



JET, composite, in chimney, #19 
Joule's experiments on tl-.«* conden- 
sation of steam, 173 
Journals of crosshead, proper clmensions 

of, 257 
— air-pump crosshead, 2S4 



LAMBETH "Water-works, engines at,, 
362; diagrams from, 86* ; duty of, 

368 
Latent heat defined, 135 

of liquefaction, 151 

— heats of steams from wa+er, alcohol, 

ether, and sulphuret of carbon, 154 
Lap of valve proper for a given amount 

of expansion, 187, 189. 190 
on eduction side, effects of, 1ST, 

193 
Lead plug, 330 

'Lem&ter,' steamer, computslk*- of fric- 
tion of, 438 
Length of pendulum to vibrate at any 

given speed, 115 
vessels should vary w'.tb intended 

speed, 421 
Leslie's explanation of the strenjr^ cS 

iron, 126 



470 



INDEX. 



LET 

Letestu, pumps by, 385 
Lever, action of the, 83 

— elbow-jointed, 89 
Levers of Stanhope press, 89 
Light, invisible, 94 

Lineal measure explained, 7 

Lines of ships, 400 ; illustrated by shape 

of fishes, 40T 
Link motion, 198 

— expansion by, diagrams showing, 355, 
356 

Liquefaction, 150 ; latent heat of, 151 ; of 

gases, 153 
Liquids, dilatation of, by heat, 143 

— specific heats of, 164 
Locomotive engines, proper proportions 

of, 301 

— boiler, example of, 329 

— efficiency of "steam vessels, 314 
Logarithms, nature of, 52 ; mode of using, 

56 
Lowest terms, how to reduce a fraction 
to, 33 



MADAGASCAK, mode of numeration 
used in, 2 
Machines, strains and strengths of, 81, 87 

— how to determine power of, 82 
Magnitude, standards of, 7 

Magnus, his experiments upon ebulli- 
tion, 16S 
Main links, 232 

— centre of land engines, 232 

— centre of marine engines, 268 

— beam of land engines, how to propor- 
tion, 233 

Maize mill, 387 

Marine engines, proportions of the parts 
of, 254-301 

— boilers, proportions of, 314 
Marquis de l'Hopital, his rule for finding 

the centrifugal force, 103 
Materials, strength of, 124 
Maudslay and Co.'s side lever engines, 

dimensions of, 290 
Maximum density of water, 139 
Midship section of ships, best form of, 

409 

— frame of ships, best position of, 417 
Mill gearing, proportions proper for, 246 
Mills: flour, 387; barley, 387; rye, 387; 

maize, 887 ; bean, 388 ; oil, 388 ; saw, 
388 ; sugar, 390 ; cotton, 391 ; weaving, 
393 ; wool, 393 ; fueling, 394 ; flax, 394 ; 
paper, 396; rolling, 397 

Millwall Ironworks, engines by, 382 

Mechanical power from" the sun, 79 

■ nature of, 90 

of the universe constant, 92 

— equivalent of heat, 91 
Melting points of solids, 14S 
Membrane pump, by Brule, 



PEE 

Mercury, relative density of, 100 

— into a vacuum, velocity of, 101 
Merry weather, pumps by, 385 
Metals, strengths of, 125 

— conducting powers of, 172 
Molecular attraction of water retards 

boiling, 168 
Momentum defined, 105 ; of rams, 105 ; 
of cannon balls, 105 

— of heavy moving bodies, how meas- 
ured, 106; of a revolving disc, 111 

indicator piston, 347 

Monitors, features of their construction, 
461 ; weak points of, 462 ; mode of de- 
stroying, 463 

Moors brought decimal system into En- 
rope, 3 

Morin's experiments on friction, 118 

Morin, General, his experiments on va- 
rious machines, 387-397 

Motive power of heat, 90 

Motion of fluids, 100 

— power required to produce, 106 

— in a circle, 107 

Multiplication, nature of, 16; multipli- 
cation table, IS, 23 ; examples of, 20 ; 
mode of performing, 22 

Multiplier defined, 20 

Multiplicand defined, 20 

Multiplication by fractions, 9 

— of fractions, 38 

' Munster,' indicator diagrams from, 340 
Mylne, his constant for velocity of water 
in pipes, 206 



NAPIEE, DAYID, his haystack boil- 
ers, 173, 316 
Numerator of fractions denned, 5 



OAK posts, proper load for, 130 
Oil mill, 388 
Ordnance, increased power of, attainable, 

459 
' Orontes 1 indicator, diagrams from, 348, 

349 
Oscillation, centre of, 114 
Oxygen required for combustion, 175 



PADDLE shaft, 294 
Paper mill, 396 
Parallel motion, how to describe the, 207 
Parallelopiped, circumscribing, 406 
Peclet's rule for proportions of chim- 
neys, 306 
Pendulum, action of the, 95 

— laws of the, 114 

— centrifugal, 116 
Percussion, centre of, 112 
Perrin, pumps by, 387 
Perry, pumps by, 387 



INDEX. 



471 



PER 

Persian wheel, 386 
Persia, 1 steamer, 3S9 
Phipps, on resistances ofbodies by, 436 
Pillars, law of strength of, 131 
Pipes, velocity of water flowing in, 199, 
433 

— and passages, proper proportions of, 
for different powers, 300 

Piston rod for land engines, 231 

— — of marine engines, 261 

— table of proportions of, 

299 

— valves, by D. Thomson, 363 
•— guns, 459 

Plates of boilers, proper thickness of, 

322 
Plus, the sign of addition, 10 
Pneumatic Despatch Company's engine, 

indicator diagrams from, 354, 355 
Portsmouth floating bridge, diagrams 

from, 355, 356 
Posts of oak, proper load for, 130 
Powers and roots of numbers, 49 
Power, m<" chanical, from the sun, 79 

— mechanical, nature of, 90 

— motive, of heat, 90 

— required to produce motion, 106 

— resident in a revolving disc, 111 

— producible by a given quantity of 
heat, 181 

in a perfect engine, 181 

— cheapest source of, 181 

— nominal, how to determine, 208; Ad- 
miralty rule for, 211 . 

of boilers an indefinite expression, 

309 

— and performance of engines, 333 

— loom weaving, 393 

— required to produce a given speed in 
steam vessels, 430, 432, 443 

Press, Stanhope, 89 

Pressure, atmospheric, how produced, 
100 

— permissible on bearings moving with 
a given speed, 121 

— strength of boiler to withstand, 322 

— safe, in a cylindrical boiler, 325, 326 

— collapsing of flues, 327 
Pressures and volumes of gas, 147 
Printing machines, 396 
Product defined, 20 

Projectiles should contain rocket com- 
position, 460 ; and have spiral feathers 
to put them into revolution, 460 

Proportion, nature of, 42 

Proportions of steam-engines, 208, 214 

engines laid down to curves, 288 

• locomotive engines, 301 

boilers, 304 

~ — wagon boilers, 310 ; of flue boilers, 
311 

Pump, combined plunger and bucket, 
363 



SAW 

Pumps, relative efficiency of different 
kinds, 387 

— by various makers, 3S7 

Pumping engine at St. Katherine's docks, 
diagram from, 346 

— engines, friction of, 367 ; duty of, 
368 



AUOTIENT defined, 24 



"DADIATIOISr of heat, 171 

It Kankine, his method of computing 
speed of steam vessels, 443 

Eatio, or Proportion, nature of, 42 

Eeaumur's thermometer, 138 

Eed-hot crucible, ice made in. 1 70 

Eeduction, 67 

Ecgnault, his experiments on dilatation 
of gases, 145 

Eegnault's formulae for the elastic forco 
of steam, 158 

Eelative bulks of water and steam at at- 
mospheric pressure, 102 

Eennie, tensile strength of metals ac- 
cording to, 126 

' Eesearch,' indicator diagram from, 
349 

Eesistance of vessels, 399 

mainly caused by friction, 423, 

442 

at bow and stern, 436 

— hydrostatic, of vessels, increases with 
speed and with breadth, 422 

'Ehone 1 steamer, proportions of engines 

and boilers of, 382 
Eichards' Indicator, 334 
Eivers, velocity of, 199 

— have water highest where stream is 
fastest, 424; effect of temperature on 
velocity, 425 

Eiveted joints, best proportions of, 320; 

strength of, 320 
Eevolving bodies, centrifugal force of, 

109 ; bursting velocity, 110 
Eocket vessels propelled by rockets, a 

new expedient of warfare, 404 
Eoman method of numeration, 3 
Eoots, square, 44 ; cube, 48 
Eopes tightened by pulling sideways, 

84 
Eule of three, 42 
Eye mill, 3S7 



SAFETY valves, rule for proportion 
ing, 219 
Saw mill, 388 ; for veneers, 389 



472 



INDEX. 



SAW 

Saw circular, 889 

— for stones. 3S9 

Screw, pressure producible by. SI 
■ differential, pressure producible by, 
81, 85 

— of Archimedes, 3S6 

1 Scud,' diagram from hot well of. 360 

Seaward and Co.'s side lever engines, di- 
mensions of, 292 

Sectional area of boiler flues or tubes, 
814 ; of chimney, 314 

Sensible heat defined, 135 

Side lever, proper proportions of, 267: 
studs of, 269 ; thickness of eve round. 
271 

engines, dimensions of, by Caird 

and Co., 2S7 ; by Maudslay, 290 ; by 
Seaward, 292 

— rods of marine engines, proper pro- 
portions of, 25S 

— rods of air-pump in marine engines, 
284 

Solid measure explained, 8 
Solids, melting points of, 148 
Specific heat defined, 135 

162; of different bodies, 163, 165, 

166 

— heats under constant pressure and 
under constant volume, 164, 167 

— gravities, tables of, 165 

of oxygen and carbonic acid, 

175 

Speed of steamers, rule for determining, 
77 

steam vessels, how to determine. 

430, 432, 443 

vessels a main condition of success 

in war, 460 

common steamers may be increas- 
ed by rocket composition, 464 

Shafts, strength of, 133 

— cf lly-wheel, 23S, 239 

— for paddles, 294 ; sizes of wrought -iron 
shafts for different powers, 294 

Ships, maximum breadth of, best posi- 
tion of, 417 

— length of, should vary with intended 
speed, 421 

— resistance of, mainly caused by fric- 
tion, 423, 442 

Spherical measure, 9 

Spheroidal condition of water, 169 

Square measure explained, 8 

— and circular inches, 9 

— roots, nature of, 44 
of fractions, 45 

— root, method of extracting, 47 
Squares and square roots, 44 

— of fractions, 45 

St. Katherine's Dock, diagram of engine 

at, 346 
Standards of magnitude, 7 
Stanhope press, levers of, 89 



Stays of boilers, 321 

Steam-engines, great waste of heat in 
91 

Steam-engine, theory of the, 134 

Steams, latent heats of, from water, al- 
cohol, ether, and sulphur of carbon 
154 

Steam and water, relative bulks of, at at- 
mospheric pressure, 102 

— of atmospheric pressure, density of, 
102 

— rushing into a vacuum, velocity o£ 
102 ; velocity the same at all pressures, 
102; velocity into the atmosphere. 
103 

— sensible and latent heat of, by M. 
Eegnault, 155; elastic force of," 155- 
161 

— expanding, mean pressure of, 285 

— ports, 216 

— pipes, proper size of, 21S 

— boilers, proportions of, 304 

— room, 315 

— ports of locomotives, 830 

— pipes of locomotives, 331 

— navigation, 899 

— vessels, locomotive efficiency of, 313 
Steamers, equation for determining 

speed of, 77 

Steamer ' Fairy, body plan of. 454 ; ' Rat- 
tler,' 45?; 'Bremen,' 456; 'Persia^ 
457 ; ' Warrior,' 458 

Stones, strength of, 125 

— machine for sawing, 339 

Strains of machines, how measured, 81, 
86 

— infinite, how produced, S9 

Stran of side rod, proper dimensions of, 

259 
connecting rod, proper dimensions 

of, 265 
Straps of air-pump side rods, 285 
Strengths of machines, how determined, 

81, S6 
Strength of main beam of an engine, 

87 
of materials, 124 ; elastic strength, 

124 
cast-iron columns. 129, 131 ; of cast 

iron beams, 133 ; of shafts, 133 
boiler to withstand any given pres 

sure, 822 
Studs of the beams of land engines, 

232 

— in side lever, 269 ; metal round studs, 
271 

Subtraction, nature of, 13; indicated by 
— or minus, 14; method of perform 
ing, 15 ; examples of, 16 

— of fractions, 34 
Sugar mill, 890 

Sun the source of mechanical power, 
79 



INDEX. 



473 



STJP 

Superficial measure explained, 7 
Superheater, proportions of, in steamer 

'Rhone, 1 SS3 
Surds or incommensurables, 46 
Surface of toiler required to evaporate a 

cubic foot of water per hour, 309 

— condensers, proportions of, in steamer 
'Hansa,' 314 

— condensers, 315 

— heating, of modem 'boilers, 375 

— condensers cause internal corrosion in 
boilers, 381 ; proportions of, in steamer 
4 Rhone,' 3S3 



TAJ3TJE of addition, 11 
Tables, multiplication, 19, 23 
Tay' steamer, dimensions of, 287 

Temperature defined, 135 

Temperatures of liquefaction and ebulli- 
tion constant, 137 

■ steam at different pressures, 159 

Tensile strengths of metals, 126 ; of 
woods, 127; crushing strengths of 
woods, 12S; iron, 129 

— strength of boiler plates, 321 

' Teviot' steamer, dimensions of, 287 

Theory of the steam-engine, 134 

Thermo-dynamics, 134 

Thermometers, 137; Centigrade, Reau- 
mur's, and Fahrenheit's compared, 
139 

Thermal unit, 162 

Thomson, D., rotative pumping engines 
by, 362; double cylinder engines by, 
862; combined plunger and. bucket 
pump by, 363 

Throttling the steam, effect of, 198 

Time during which bodies have fallen 
determinable from their velocity, 09 

~by height fallen 

through, 98 

Toothed wheels, proportions proper for, 
246 

Torsion, strength to resist, of different 
metals, 133 

Transverse section of ships, best form of, 
409 

Tubes of locomotive boilers, S30 

'Tweed' steamer, dimensions of, 287 

Tylor, pumps by, 385 



'TTLSTER, 1 indicator diagrams from, 

U 342, 345, 352 
Unit, meaning of the term, 5 
— of heat, 162 
Uptake of boilers, sectional area of, 376 



\rACUUM, velocity of air, water, and 
mercury into. 101 ; of steam and 
gas, 102 



WAG 

Values of different coals in generating 

steam, 177 
Valve piston, by D. Thomson, 363 
Vaporisation, 152 ; latent heat of, 154 
Vapours and gases, difference between 

152 
Velocities, virtual, law of, 79 
Velocity of falling bodies, 95 
determinable from height 

fallen, 97 ; from time of falling, 97 
air, water, and mercury into a va« 

cuum, 101 ; of steam and gas, 102 
rotation that will burst by centrifa* 

gal force, 110 

— permissible in bearings moving un- 
der a given pressure, 123 

water in rivers, canals, and pipes* 

199 
water flowing in pipes and canals. 

433 
Veneer saw, 389 
Vermicelli machine, 388 
Vertical tubes, advantages of, 377 
Vessels, resistance of, 899 ; proper shape 

of, 401 

— maximum breadth of, best position 
of, 417 

— length of, should vary with intended 
speed, 421 

— resistance of, mainly caused by fric- 
tion, 423, 442 

Vibrations of pendulums, rule for deter 

mining, 115 
' Victoria and Albert,' indicator diagram 

from, 352 
Virtual velocities, law of, 79 
Viscosity or molecular attraction, 163 
Vis viva, nature of, 90 
Volumes, relative, of water and steam a* 

atmospheric presstire, 102 

— and pressures of gases, 146 

— of gases, 166 

Vulgar fractions, nature of, 5 



TTTAGOlSr boilers, proportions of, 310 
VY Water, relative density of, 100 

— into a vacuum, velocity of, 101 

— and steam, relative buiks of, at atmos 
pheric pressure, 102 

— maximum density of, 139 

— weight of, at 32°, 139 

— velocity of, in rivers, canals, and 
pipes, 433 

— works, indicator diagram from pump, 
361 

— lines of ships, 400 ; illustrated by 
shape of fishes, 408 

— in pipes, friction of the same at all 
pressures, 429 

— velocity of, in pipes, 434; in canalSj 
1 434 



474 



INDEX. 



WAR 

War, maritime, new resources available 
for, 461-464 

'Warrior' steamer, body plan of, 458; 
transverse section of, 459 

Waste water pipe, to find the proper di- 
ameter of, 2'24 

Wave .raised by a vessel, 414, 419 ; mo- 
tion of, 420 

Weaving by steam, 393 ; by compressed 
air, 39S 

Weights lifted by machines, 82 

Weahani , s double cylinder engine, 36S 



ZUB 

Wheels, teeth of, 247 
Winch weights lifted by, 81 
Wirtz's Zuric/' machine, 386 
Woods, strength of, 125 
Wool-spinning mill, 393 
Working beam of land engines, how ts 
proportion, 233 



ZERO, absolute, 137 
Zurich machine, 336 



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are glad that the translation has fallen into such good hands as those of Professor 
Everett. . . . It will form an admirable text-book. 11 

Nature. 

"The engravings with which the work is illustrated are especially good, a point in 
which most of our English scientific works are lamentably deficient. The clearness 
of Deschaners explanations is admirably preserved in the translation, while the value 
ot the tratise is considerably enhanced by some important additions. . . . We believe 
the book will be found to supply a real need." 

D. APPLETON & CO., New York. 



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